An ellipse is the set of all points (
In this article, we will cover the concept of the Diameter of the Ellipse. This category falls under the broader category of Coordinate Geometry, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. A total of seven questions have been asked on JEE MAINS( 2013 to 2023) from this topic.
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What is the Diameter of the Ellipse?
The locus of the mid-points of a system of parallel chords of an ellipse is called a diameter and the point where the diameter intersects the ellipse is called the vertex of the diameter.
Locus of Mid Point
Let
Hence, the locus of the mid-point is
Two diameters are said to be conjugate when each bisects all chords parallel to the other.
If
If
1) The tangent at the extremity of any diameter is parallel to the chords it bisects or parallel to the conjugate diameter.
2) The tangents at the ends of any chord meet on the diameter which bisects the chord.
Properties of conjugate diameters
1) The eccentric angles of the ends of a pair of conjugate diameters of an ellipse differ by a right angle.
2) The sum of the square of any two conjugate semi-diameters of an ellipse is constant and equal to the sum of squares of the semi-axis.
3) The product of the focal distances of a point on an ellipse is equal to the square of the semi-diameter which is conjugate to the diameter through the point.
4) Two conjugate diameters are called equi conjugate if their lengths are equal.
Example 1: The Locus of midpoints of chords of the ellipse
Solution:
Locus of the mid-point of the chord of the ellipse
The equation of a tangent to the ellipse
from eq (i) and eq (ii)
Hence, the correct answer is
Example 2: If the product of focal distances of a point
D is
By using the definition of an ellipse,
Or
But
Hence, the correct answer is 1
Example 3: If one end of the diameter of the ellipse
Solution: Since every diameter of an ellipse passes through the centre and is bisected by it, therefore the coordinates of the other end are
Hence, the answer is
Example 4: A ray emanating from the point
Solution: For point P y-coordinate
Given ellipse is
co-ordinate of co-ordinate of P is
Equation of reflected ray
(i.e.PS) is
Hence, the correct answer is
Example 5: If the points of intersection of the ellipses
Solution: Subtracting in order to find their points of intersection, we get
The above equation will represent a pair of conjugate diameters of the first ellipse if
But
Hence, the correct answer is 2
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