The locus of a point moves in a plane such that the ratio of the distance from a fixed point (focus) to the distance from a fixed line (directrix) is constant. The line perpendicular to the tangent to the curve at the point of contact is normal to the Hyperbola. In real life, we use Hyperbolas in race tracks, architectural design, mirrors, and celestial orbits.
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In this article, we will cover the concept of Equation of Normal in Point Form and Parametric Form. 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 fifteen questions have been asked on JEE MAINS( 2013 to 2023) from this topic including two in 2018, one in 2020, one in 2021, four in 2022, and one in 2023.
A Hyperbola is the set of all points (
OR,
The locus of a point moves in a plane such that the ratio of the distance from a fixed point (focus) to the distance from a fixed line (directrix) is constant. The constant is known as eccentricity e and for hyperbola e 1.
Equation of Normal of Hyperbola in Point form
The equation of normal at
We know that the equation of tangent in point form at
Slope of tangent at
Hence, the equation of normal at point
Equation of Normal of Hyperbola in Parametric form
The equation of normal at
The equation of normal in point form is
The equation of normal at
The equation of normal in point form is
Equation of Normal of Hyperbola in Slope form
The equation of normal of slope m to the hyperbola,
The equation of normal at
Let,
Hence, the equation of normal becomes
Pair of Tangents
The combined equation of pair of tangents from the point
Note:
The formula
Chord of Contact
The equation of chord of contact of tangents from the point
i.e.
Equation of Chord bisected at a given point
The equation of chord of the hyperbola
is
or,
Example 1: Let
Solution: Equation of tangent to the hyperbola
passing through
Equation of tangent with positive slopes
Intersection points:
Hence, the answer is 8.
Example 2: Consider a hyperbola
Solution
equation of the tangent at
tangent meet
latus rectum
Area of
Hence, the answer is
Example 3: The vertices of a hyperbola
Solution
The equation of normal is
The equation of normal is
Hence, the answer is 216
The normal to a hyperbola is a perpendicular line at any point of tangency. The normal to hyperbola helps us with the various problems of calculus. Understanding the normal of a hyperbola is essential in geometry and calculus for analyzing the geometric properties and interactions of parabolic curves with straight lines.
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