Question

The equation of the chord of the hyperbola $25x^2 - 16y^2 = 400$ that is bisected at point (5, 3) is :

• A. $135x - 48y = 481$
• B. $125x - 48y = 481$
• C. $125x - 4y = 48$
• D. None of these

Answer 1

Answer: (B)

$125x - 48y = 481$

Answer 2

Let the equation of the hyperbola be $S: 25x^2 - 16y^2 = 400$. The point of bisection of the chord is $(x_1, y_1) = (5, 3)$.

The equation of the chord of a conic section $S=0$ that is bisected at the point $(x_1, y_1)$ is given by the formula $T = S_1$. Here, $S = 25x^2 - 16y^2 - 400$.

$T$ is obtained by replacing $x^2$ with $xx_1$ and $y^2$ with $yy_1$ in the terms involving $x^2$ and $y^2$, and keeping the constant term as is. $T = 25xx_1 - 16yy_1 - 400$.

$S_1$ is obtained by substituting the coordinates $(x_1, y_1)$ into the equation $S$. $S_1 = 25x_1^2 - 16y_1^2 - 400$.

The equation of the chord is $T = S_1$: $25xx_1 - 16yy_1 - 400 = 25x_1^2 - 16y_1^2 - 400$ $25xx_1 - 16yy_1 = 25x_1^2 - 16y_1^2$

Substitute the given point $(x_1, y_1) = (5, 3)$: $25x(5) - 16y(3) = 25(5)^2 - 16(3)^2$ $125x - 48y = 25(25) - 16(9)$ $125x - 48y = 625 - 144$ $125x - 48y = 481$