Position of Two Points with Respect to a Line

Q: What does "position of two points with respect to a line" mean in geometry?

This concept refers to determining whether two given points lie on the same side of a line, on opposite sides of the line, or if one or both points lie directly on the line. It helps us understand the spatial relationship between points and lines in a coordinate plane.

Q: How can we determine if two points are on the same side of a line?

To determine if two points are on the same side of a line, we can use the line equation ax + by + c = 0. Substitute the coordinates of both points into the left side of the equation. If the results have the same sign (both positive or both negative), the points are on the same side of the line.

Q: What does it mean when two points are on opposite sides of a line?

When two points are on opposite sides of a line, it means that a line segment connecting these two points will intersect the given line. Mathematically, when we substitute the coordinates of the points into the line equation, the results will have opposite signs (one positive and one negative).

Q: How do we know if a point lies exactly on a line?

A point lies exactly on a line if its coordinates satisfy the equation of the line. When we substitute the point's coordinates (x, y) into the line equation ax + by + c = 0, the left side should equal zero. This means the point satisfies the line equation and therefore lies on the line.

Q: What is the significance of the sign when substituting point coordinates into a line equation?

The sign obtained when substituting point coordinates into a line equation indicates the point's position relative to the line. A positive result means the point is on one side of the line, a negative result means it's on the other side, and a zero result means the point is on the line itself.

Q: How does the concept of "position of points" extend to three-dimensional space?

In 3D space, we consider the position of points with respect to a plane rather than a line. The principle is similar: we use the equation of the plane to determine if points are on one side, the other side, or on the plane itself. The concept becomes more complex as we now have three possible dimensions of variation.

Q: Can a point be equidistant from three non-collinear points?

Yes, a point can be equidistant from three non-collinear points. This point is the center of the circle that passes through all three points (the circumcenter of the triangle formed by the three points). In terms of line positions, this point is located at the intersection of the perpendicular bisectors of the sides of the triangle.

Q: How can we use the position of points to determine if a point is inside or outside a circle?

While a circle is not a line, we can use the concept of point position relative to the circle's equation. If we have a circle with equation (x - h)² + (y - k)² = r², a point (x, y) is inside the circle if (x - h)² + (y - k)² r². This is analogous to how we use inequalities with lines.

Position of Two Points with Respect to a Line

Edited By Komal Miglani | Updated on Jul 02, 2025 07:45 PM IST

The position of two points concerning a line is a fundamental concept in geometry and analytical mathematics. It has many applications used in real life from physics, robotics, etc. The position of the points on the line determines whether the position is on the line, same side of the line, or on the opposite side.

Position of Two Points with Respect to a Line

Position of two points with respect to a line

Two given points $\mathrm{A}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ and $\mathrm{B}\left(\mathrm{x}_2, \mathrm{y}_2\right)$ lies on the same side of a line $a x+b y+c=0$ when $\frac{\mathrm{ax}_1+\mathrm{by}_1+\mathrm{c}}{\mathrm{ax}_2+\mathrm{by}_2+\mathrm{c}}>0$ and points lie on the opposite side when $\frac{\mathrm{ax}_1+\mathrm{by}_1+\mathrm{c}}{\mathrm{ax}_2+\mathrm{by}_2+\mathrm{c}}<0$.

Note:

The side of the line where origin lies is known as the origin side.
A point $(p, q)$ will lie on the origin side of the line $a x+b y+c=0$ if $\frac{a p+b q+c}{a .0+b .0+c}>0$, meaning ap $+\mathrm{bq}+\mathrm{c}$ and c will have the same sign.
A point $(p, q)$ will lie on the non-origin side of the line $a x+b y+c$ $=0$, if $\frac{a p+b q+c}{a .0+b .0+c}<0$, meaning $\mathrm{ap}+\mathrm{bq}+\mathrm{c}$ and c will have the opposite sign.

Position of a point which lies inside a triangle

Let $P\left(x_1, y_1\right)$ be the point that lies inside the triangle

The equations of sides of a triangle are

$\begin{aligned} & A B: a_1 x+b_1 y+c_1=0 \\ & B C: a_2 x+b_2 y+c_2=0 \\ & C A: a_3 x+b_3 y+c_3=0\end{aligned}$

First find the coordinates of vertices of triangle ABC

Let $A=\left(x^{\prime}, y^{\prime}\right), \quad B=\left(x^{\prime \prime}, y^{\prime \prime}\right)$ and $C=\left(x^{\prime \prime \prime}, y^{\prime \prime \prime}\right)$

And if coordinates of vertices of triangle ABC is given then find equation of sides of triangle ABC.

If point P lies inside the triangle, then P and A must be same side of BC, P and B must be same side of AC and P and C must be same side of AB, then.

Solved Examples Based on Position of Two Points with Respect to a Line:

Example 1: if $P\left(p, p^2\right)$ lies inside the triangle having sides along the lines $2 x+3 y=1, x+2 y-3=0,6 y=5 x-1$ then the value of p?
1) $p \epsilon(1 / 2,1) \cup(-3 / 2,-1)$
2) $p \epsilon(1 / 2,1) \cup(-2,-1)$
3) $p \epsilon(1,2) \cup(-3 / 2,-1)$
4) None of these

Solution

First find the coordinates of vertices of triangle ABC

Let $A=\left(x^{\prime}, y^{\prime}\right), \quad B=\left(x^{\prime \prime}, y^{\prime \prime}\right)$ and $C=\left(x^{\prime \prime \prime}, y^{\prime \prime \prime}\right)$

And if coordinates of vertices of triangle ABC is given then find equation of sides of triangle ABC.

If point P lies inside the triangle, then P and A must be same side of BC, P and B must be same side of AC and P and C must be same side of AB, then.

$\begin{aligned} & \frac{a_2 x_1+b_2 y_1+c_2}{a_2 x^{\prime}+b_2 y^{\prime}+c_2}>0 \\ & \frac{a_3 x_1+b_3 y_1+c_3}{a_3 x^{\prime \prime}+b_3 y^{\prime \prime}+c_3}>0 \\ & \frac{a_1 x_1+b_1 y_1+c_1}{a_1 x^{\prime \prime \prime}+b_1 y^{\prime} 1^{\prime}+c_3}>0 \\ & 2 x+3 y=1 \ldots(i) \\ & x+2 y=3 \ldots(\text { ii }) \\ & 5 x-6 y=1 . .(i i i) \\ & \text { equation (i)-2*equation(ii) } \\ & \text { PointA(-7, 5) } \\ & 5^* \text { equation (ii)-equation(iii) } \\ & \text { pointB(5/4,7/8) } \\ & 5^* \text { equation(i)-2*equation(iii) } \\ & \text { Point } C(1 / 3,1 / 9)\end{aligned}$

So $\mathrm{A}, \mathrm{B}, \mathrm{C}$ be vertices of the triangle.
$
\begin{aligned}
& \mathrm{A} \equiv(-7,5), \mathrm{B} \equiv(5 / 4,7 / 8) \\
& \mathrm{C} \equiv(1 / 3,1 / 9)
\end{aligned}
$

If P lies in-side the $\triangle \mathrm{ABC}$, then sign of P will be the same as sign of a w.r.t. the line BC
$
\Rightarrow \quad 5 p-6 p^2-1<0 \Rightarrow\left(-\infty, \frac{1}{3}\right) U\left(\frac{1}{2}, \infty\right)
$

Similarly $2 p+3 p^2-1>0 \Rightarrow(-\infty,-1) U\left(\frac{1}{3}, \infty\right)$
And, $\quad p+2 p^2-3<0 \Rightarrow\left(\frac{-3}{2}, 1\right)$.
Solving, (1), (2) and (3) for $p$ and then taking intersection, We get $p \in(1 / 2,1) \cup(-3 / 2,-1)$.

Example 3: Let the point $\mathrm{P}(\alpha, \beta)$ be at a unit distance from each of the two lines $\mathrm{L}_1: 3 \mathrm{x}-4 \mathrm{y}+12=0$, and $\mathrm{L}_2: 8 \mathrm{x}+6 \mathrm{y}+11=0$. If P lies below $\mathrm{L}_1$ and above $\mathrm{L}_2$, then $100(\alpha+\beta)$ is equal to
1) -14
2) 42
3) -22
4) 14

Solution

$
\begin{aligned}
& \mathrm{P} \text { and origin lie on same side of } \mathrm{L}_1 \\
& \Rightarrow \frac{3 \alpha-4 \beta+12}{12}>0 \\
& \Rightarrow 3 \alpha-4 \beta+12>0
\end{aligned}
$

Similarly for $\mathrm{L}_2$
$
\begin{aligned}
& \frac{8 \alpha+6 \beta+11}{11}>0 \\
& \Rightarrow 8 \alpha+6 \beta+11>0
\end{aligned}
$

Also distance from $\mathrm{L}_1$ and $\mathrm{L}_2=1$
$
\begin{aligned}
& \Rightarrow \frac{|3 \alpha-4 \beta+12|}{5}=1 \\
& \Rightarrow 3 \alpha-4 \beta+12=5 \quad(\text { using (ii)) } \\
& \text { And } \frac{|8 \alpha+6 \beta+11|}{10}=1 \\
& 8 \alpha+6 \beta+11=10
\end{aligned}
$

Solving these 2 equations
$
\begin{aligned}
& \alpha=\frac{-23}{25}, \quad \beta=\frac{53}{50} \\
& \therefore 100\left(\alpha_1 \beta\right)=14
\end{aligned}
$

Hence, the correct option is 4.

Example 4: Find the range of $\theta$ in the interval $(0, \pi)$ such that the points $(3,5)$ and $(\sin \theta, \cos \theta)$ lie on the same side of the line $\mathrm{x}+\mathrm{y}-1=0$.
1) $0<\theta<\pi / 6$
2) $0<\theta<3 \pi / 4$
3) $\frac{\pi}{2}<\theta<\pi$
4) $0<\theta<\pi / 2$

Solution
$
\begin{aligned}
& 3+5-1=7>0 \\
& \therefore \sin \theta+\cos \theta-1>0 \\
& \Rightarrow \sin (\pi / 4+\theta)>1 / \sqrt{ } 2 \Rightarrow \pi / 4<\pi / 4+\theta<3 \pi / 4 \Rightarrow 0<\theta<\pi / 2
\end{aligned}
$

Hence, the answer is the option (4).

Example 5: Find the range of $\theta$ in the interval $(0, \pi)$ such that the points $(3,5)$ and $(\sin \theta, \cos \theta)$ lie on the same side of the line $\mathrm{x}+\mathrm{y}-1=0$.
1) $0<\theta<\frac{\pi}{6}$
2) $0<\theta<3 \frac{\pi}{4}$
3) $\pi<\theta<\pi$
4) $0<\theta<\frac{\pi}{2}$

Solution:
$
\begin{aligned}
& 3+5-1=7>0 \\
& \therefore \sin \theta+\cos \theta-1>0
\end{aligned}
$
$\Rightarrow \sin (\pi / 4+\theta)>1 / \sqrt{ } 2 \Rightarrow \pi / 4<\pi / 4+\theta<3 \pi / 4 \Rightarrow 0<\theta<\pi / 2$.Hence, the answer is the option (4)

Frequently Asked Questions (FAQs)

1. What does "position of two points with respect to a line" mean in geometry?

This concept refers to determining whether two given points lie on the same side of a line, on opposite sides of the line, or if one or both points lie directly on the line. It helps us understand the spatial relationship between points and lines in a coordinate plane.

2. How can we determine if two points are on the same side of a line?

To determine if two points are on the same side of a line, we can use the line equation ax + by + c = 0. Substitute the coordinates of both points into the left side of the equation. If the results have the same sign (both positive or both negative), the points are on the same side of the line.

3. What does it mean when two points are on opposite sides of a line?

When two points are on opposite sides of a line, it means that a line segment connecting these two points will intersect the given line. Mathematically, when we substitute the coordinates of the points into the line equation, the results will have opposite signs (one positive and one negative).

4. How do we know if a point lies exactly on a line?

A point lies exactly on a line if its coordinates satisfy the equation of the line. When we substitute the point's coordinates (x, y) into the line equation ax + by + c = 0, the left side should equal zero. This means the point satisfies the line equation and therefore lies on the line.

5. What is the significance of the sign when substituting point coordinates into a line equation?

The sign obtained when substituting point coordinates into a line equation indicates the point's position relative to the line. A positive result means the point is on one side of the line, a negative result means it's on the other side, and a zero result means the point is on the line itself.

6. How does the concept of "position of points" extend to three-dimensional space?

In 3D space, we consider the position of points with respect to a plane rather than a line. The principle is similar: we use the equation of the plane to determine if points are on one side, the other side, or on the plane itself. The concept becomes more complex as we now have three possible dimensions of variation.

7. Can a point be equidistant from three non-collinear points?

Yes, a point can be equidistant from three non-collinear points. This point is the center of the circle that passes through all three points (the circumcenter of the triangle formed by the three points). In terms of line positions, this point is located at the intersection of the perpendicular bisectors of the sides of the triangle.

8. How can we use the position of points to determine if a point is inside or outside a circle?

While a circle is not a line, we can use the concept of point position relative to the circle's equation. If we have a circle with equation (x - h)² + (y - k)² = r², a point (x, y) is inside the circle if (x - h)² + (y - k)² < r², on the circle if (x - h)² + (y - k)² = r², and outside if (x - h)² + (y - k)² > r². This is analogous to how we use inequalities with lines.

9. How does the concept of "position of points" extend to conic sections other than lines?

For conic sections like parabolas, ellipses, and hyperbolas, we can still talk about the position of points relative to the curve, though it's more complex than with lines. Instead of "sides," we often refer to the interior and exterior of the conic. For a parabola, points can be inside the parabola (on the same side as the vertex), on the parabola, or outside. For ellipses, points can be inside, on, or outside the ellipse. For hyperbolas, we have points inside either branch, on either branch, or outside both branches.

10. Can three points be on the same side of a line?

Yes, three or more points can be on the same side of a line. If we substitute the coordinates of all these points into the line equation and get results with the same sign (all positive or all negative), then all the points are on the same side of the line.

11. What is the relationship between the position of points and the slope of a line?

The slope of a line doesn't directly determine the position of points with respect to the line. However, the slope is part of the line equation, which we use to determine point positions. The slope affects how the line divides the plane, but points can be on either side regardless of the slope's value.

12. How does the concept of "position of points" relate to parallel lines?

For parallel lines, the relative position of points remains consistent. If two points are on the same side of one line, they will also be on the same side of any line parallel to it. This is because parallel lines never intersect and maintain a constant distance from each other.

13. What role does the y-intercept play in determining the position of points?

The y-intercept (c in the equation ax + by + c = 0) affects where the line crosses the y-axis, which in turn influences how the line divides the plane. Changing the y-intercept shifts the line vertically, potentially altering which side of the line certain points fall on.

14. How can we visualize the "sides" of a line in a coordinate plane?

We can visualize the sides of a line by imagining the line dividing the coordinate plane into two regions. One region can be considered "above" or "to the left" of the line (typically associated with positive results when substituting into the line equation), while the other region is "below" or "to the right" (typically associated with negative results).

15. What happens to the position of points when a line is rotated around a fixed point?

When a line is rotated around a fixed point, the positions of other points relative to the line may change. Points that were on one side of the line might end up on the other side after rotation, or points that were on the line might move off it. The fixed point, however, will always remain on the line.

16. How does the concept of "position of points" relate to linear inequalities?

Linear inequalities use the concept of point position to define regions in the coordinate plane. For example, ax + by + c > 0 represents all points on one side of the line ax + by + c = 0, while ax + by + c < 0 represents all points on the other side. The line itself is included or excluded based on whether the inequality is strict (> or <) or inclusive (≥ or ≤).

17. Can a point be equidistant from two parallel lines?

Yes, a point can be equidistant from two parallel lines. In fact, all points on a line that is parallel to and midway between the two given lines will be equidistant from both lines. This midway line is called the midline or axis of symmetry between the parallel lines.

18. How does the distance formula relate to determining the position of points with respect to a line?

While the distance formula isn't directly used to determine which side of a line a point is on, it can be used to calculate the perpendicular distance from a point to a line. This distance can then be used to determine which of two points is closer to the line, or to find points that are equidistant from the line.

19. What is the significance of the normal vector in determining point positions?

The normal vector of a line (a vector perpendicular to the line) can be used to determine which side of the line a point is on. If we consider the normal vector pointing to one side of the line, points on that side will have a positive dot product with the normal vector, while points on the other side will have a negative dot product.

20. What is the relationship between the position of points and the angle between two lines?

The angle between two lines doesn't directly determine the position of points, but it affects how the lines divide the plane. As the angle between lines changes, points may shift from being on the same side of both lines to being on different sides. When lines are perpendicular, they divide the plane into four quadrants, each containing points with a unique combination of positions relative to the two lines.

21. How can we use vector methods to determine the position of points with respect to a line?

We can use vector methods by considering the line as a directed line segment. If we have a vector v along the line and a vector w from a point on the line to the point we're checking, we can use the cross product v × w. The sign of the z-component of this cross product will indicate which side of the line the point is on.

22. What is the importance of understanding point positions in computer graphics?

In computer graphics, understanding point positions relative to lines (or planes in 3D) is crucial for tasks like clipping (determining which parts of an object are visible), collision detection (determining if objects intersect), and rendering (deciding which pixels to color). It's a fundamental concept in creating and manipulating digital images and 3D models.

23. How does the concept of "position of points" relate to the idea of a half-plane?

A half-plane is precisely defined by the position of points with respect to a line. One half-plane consists of all points on one side of the line, while the other half-plane consists of all points on the other side. The line itself may be included in one of the half-planes or neither, depending on how we define our half-planes (open or closed).

24. Can a line segment intersect a line if its endpoints are on the same side of the line?

No, if both endpoints of a line segment are on the same side of a line, the line segment cannot intersect that line. For a line segment to intersect a line, its endpoints must be on opposite sides of the line (or at least one endpoint must be on the line itself).

25. How does changing the coefficients in a line equation (ax + by + c = 0) affect the position of points?

Changing the coefficients a and b in the line equation changes the slope of the line, which can dramatically alter which points are on which side. Changing c (the y-intercept) shifts the line vertically, which can also change point positions. However, points that were on the same side will remain on the same side if only c is changed.

26. What is the relationship between the position of points and the area of a triangle?

The position of points with respect to a line is crucial in calculating the area of a triangle. If we consider one side of the triangle as our reference line, the area of the triangle is directly related to the perpendicular distance of the third point from this line. This forms the basis of the formula: Area = (1/2) * base * height.

27. How can we use the concept of point position to determine if three points are collinear?

Three points are collinear if they all lie on the same straight line. We can check this by using two of the points to define a line, and then determining if the third point lies exactly on this line. If it does (i.e., if substituting its coordinates into the line equation yields zero), then the points are collinear.

28. What is the significance of the position of points in the context of linear programming?

In linear programming, the position of points with respect to constraint lines is crucial. The feasible region of a linear programming problem is defined by points that satisfy all constraints simultaneously - in other words, points that are on the correct side of all constraint lines. The optimal solution is often found at the intersection of these constraint lines.

29. How does the concept of "position of points" relate to the idea of convexity in geometry?

Convexity is closely related to the position of points. A set is convex if, for any two points in the set, all points on the line segment connecting them are also in the set. This means that for any line drawn through a convex set, all points of the set must be on one side of the line or on the line itself - they can't be on both sides.

30. Can a point be on one side of a line in one coordinate system but on the other side in a different coordinate system?

No, the relative position of a point with respect to a line is invariant under coordinate transformations. While the specific equation of the line might change in different coordinate systems, the fundamental geometric relationship between the point and the line remains the same. Points that are on one side of the line will remain on that same side regardless of the coordinate system used.

31. How does the concept of "position of points" relate to the perpendicular distance from a point to a line?

The perpendicular distance from a point to a line is the shortest distance from the point to any point on the line. This distance is always positive, regardless of which side of the line the point is on. However, we can assign a sign to this distance based on which side of the line the point is on, creating a "signed distance" that indicates both the distance and the position relative to the line.

32. What is the relationship between the position of points and the concept of a line of symmetry?

A line of symmetry divides a figure into two identical halves that are mirror images of each other. In terms of point positions, for every point on one side of the line of symmetry, there exists a corresponding point on the other side that is equidistant from the line. The line of symmetry itself contains points that are their own mirror images.

33. How can we use the position of points to determine if two line segments intersect?

To determine if two line segments intersect, we can check the position of the endpoints of each segment with respect to the line containing the other segment. If the endpoints of one segment are on opposite sides of the line containing the other segment, and vice versa, then the segments intersect. If all endpoints are on the same side of either line, the segments do not intersect.

34. What is the significance of the position of points in the context of geometric transformations?

In geometric transformations like rotations, reflections, and translations, the position of points relative to certain lines can be crucial. For example, in a reflection, points change sides with respect to the line of reflection. In a rotation, points change their position relative to all lines except those passing through the center of rotation. Understanding these relationships helps in predicting and calculating the results of transformations.

35. How does the concept of "position of points" relate to the idea of a tangent line?

A tangent line to a curve at a point touches the curve at that point but does not cross it. In terms of point positions, this means that all points on the curve near the point of tangency (except the point itself) are on the same side of the tangent line. This property distinguishes tangent lines from secant lines, which do cross the curve.

36. How does the position of points relate to the concept of a polygon's interior and exterior?

The interior and exterior of a polygon are defined by the position of points with respect to the polygon's edges (which are lines). A point is in the interior of a convex polygon if it's on the same side of all the polygon's edges. It's in the exterior if it's on the opposite side of at least one edge. Points on the edges themselves are neither interior nor exterior but form the polygon's boundary.

37. What is the relationship between the position of points and the concept of a line's direction vector?

A line's direction vector points along the line and is parallel to it. While it doesn't directly determine which side of the line a point is on, it can be used in conjunction with a point on the line to form a vector. The cross product of this vector with the vector from the point on the line to any other point will have a direction (up or down in 2D) that indicates which side of the line the point is on.

38. What is the significance of the position of points in the context of vector projection?

In vector projection, we project one vector onto another, effectively finding the component of one vector that's parallel to another. The position of the endpoint of the projected vector relative to the line of the projection vector can tell us about the angle between the original vectors. If the projected point is on the same side as the projection vector's endpoint, the angle is acute; if it's on the opposite side, the angle is obtuse.

39. How does the concept of "position of points" relate to the idea of a support line in convex geometry?

A support line of a convex set is a line that touches the set at one or more points, with the entire set lying on one side of the line. This concept directly uses the idea of point position: all points in the set must be on or to one side of the support line. Support lines are important in defining the shape and properties of convex sets.

40. Can the position of points with respect to a line be used to define an ordering of points?

Yes, the position of points with respect to a line can define a partial ordering of points. For example, we could order points based on their signed perpendicular distance from the line. This would create an ordering where points on one side of the line are "less than" points on the line, which are "less than" points on the other side. However, this ordering wouldn't distinguish between points that are equidistant from the line.

41. What is the relationship between the position of points and the concept of a separating line?

A separating line is a line that completely separates two sets of