If a hyperbola passes through the point P10, 16, and it has vertices at ±6, 0, then the equation of the normal to it at P, is.

Correct answer: (B) 2x+5y=100. Let the equation of hyperbola is x^2a^2-y^2b^2=1. As, vertex is at ±6,0, and it passes through 10, 16, So, the equation of hyperbola is x^236-y^2144=1. Equation of normal to the hyperbola at x_1, y_1 is a^2xx_1+b^2yy_1=a^2+b^2 Thus, on putting values, equation of normal is 2x+5y=100.

Q32 / 75 · JEE Main 2020 (08 Jan Shift 2) · Mathematics

If a hyperbola passes through the point $P (10, 16)$ , and it has vertices at $(\pm 6, 0)$ , then the equation of the normal to it at $P$ , is.

A
$3 x + 4 y = 94$
Correct answer
B
$2 x + 5 y = 100$
C
$x + 2 y = 42$
D
$x + 3 y = 58$

Solution

Let the equation of hyperbola is $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ .

As, vertex is at $(\pm 6, 0)$ , and it passes through $(10, 16)$ ,
So, the equation of hyperbola is $\frac{x^{2}}{36} - \frac{y^{2}}{144} = 1$ .
Equation of normal to the hyperbola at $(x_{1}, y_{1})$ is $\frac{a^{2} x}{x_{1}} + \frac{b^{2} y}{y_{1}} = a^{2} + b^{2}$

Thus, on putting values, equation of normal is $2 x + 5 y = 100$ .

Hence, the correct answer is option: 'B'.

Relevant Questions

Let the tangent to the parabola y 2 = 12 x at the point 3 , be perpendicular to the line 2 x + 2 y = 3 . Then the square of distance of the 11 Apr, 2023 (E)Normal to hyperbola
If the maximum distance of normal to the ellipse x 2 4 + y 2 b 2 = 1 , b 2 , from the origin is 1 , then the eccentricity of the ellipse is:31 Jan, 2023 (M)Normal to hyperbola
The vertices of a hyperbola H are 6 , 0 and its eccentricity is 5 2 . Let N be the normal to H at a point in the first quadrant and parallel25 Jan, 2023 (M)Normal to hyperbola
Let the tangent drawn to the parabola y 2 = 24 x at the point , is perpendicular to the line 2 x + 2 y = 5 . Then the normal to the hyperbol26 Jul, 2022 (M)Normal to hyperbola
The normal to the hyperbola x 2 a 2 - y 2 9 = 1 at the point 8 , 3 3 on it passes through the point26 Jun, 2022 (E)Normal to hyperbola
Let the eccentricity of the hyperbola x 2 a 2 - y 2 b 2 = 1 be 5 4 . If the equation of the normal at the point 8 5 , 12 5 on the hyperbola 25 Jun, 2022 (E)Normal to hyperbola