Question

If chord of contact of tangents from two points $(x_1,y_1)$ and $(x_2,y_2)$ to the ellipse $\frac{x^2}{52}+\frac{y^2}{13}=1$ are at right angles, then ratio of the product of abscissas and ordinates is

• A. $-16:1$
• B. $4:1$
• C. $16:1$
• D. None of these

Answer 1

The correct option is A $-16:1$

Equation of tangent from $(x_1,y_1)$ to ellipse $\frac{x^2}{52}+\frac{y^2}{13}=1$ is $\frac{{xx}_1}{52}+\frac{{yy}_1}{13}=1$

slope of this tangent, $m_1$ = $\frac{{-b}^2}{a^2}\times \frac{x_1}{y_1}$

$m_1=\frac{{-x}_1}{{4y}_1}$

slope of tangent passing through $(x_2,y_2)$ to ellipse $\frac{x^2}{52}+\frac{y^2}{13}=1$ is

$m_2=\frac{{-b}^2}{a^2}\times \frac{x_2}{y_2}$

$=\frac{{-x}_2}{{4y}_2}$

As they make angle of ${90}^0$

$m_1\times m_2=-1$

$\frac{{-x}_1}{{4y}_1}\times \frac{{-x}_2}{{4y}_2}=-1$

$\frac{x_1{x}_2}{y_1{y}_2}=\frac{-16}{1}$

Ratio of product of abscissas and ordinates is $-16:1$