Question

What is the equation of chord of contact of tangents drawn from $P(10,8)$ to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1.$

• A. $4x-5y=5$
• B. $5x-4y=10$
• C. $4x+5y=10$
• D. $4x-5y=10$

Answer 1

The correct option is C $4x+5y=10$

We have to find the equation of chord of contact for a point given. We can find it by two ways. Using formula and comparing equations.
The equation of chord of contact of a second degree curve is given by
$T=0\Rightarrow \frac{10x}{25}+\frac{8y}{16}=1\Rightarrow \frac{2x}{5}+\frac{y}{2}=1\Rightarrow 4x+5y=10$

In the second method we will write the equation of two tangents assuming two points of contact, $(x_1,y_1)$ and$(x_2,y_2)$. The equations of tangents at these points are

$\Rightarrow \frac{x{x}_1}{25}+\frac{y{y}_1}{16}=1\text{and}\frac{x{x}_2}{25}+\frac{y{y}_2}{16}=1$

Since these two lines pass through $(10,8),$
we have, $\frac{10x_1}{25}+\frac{8y_1}{16}=1\text{and}\frac{10x_2}{25}+\frac{8y_2}{16}=1$

Comparing these two equations, we find that an equation of the form
$\frac{10x}{25}+\frac{8y}{16}=1$
will be satisfied by both $(x_1,y_1)$ and $(x_2,y_2).$
So the equation of line passing through these two points will be $\frac{10x}{25}+\frac{8y}{16}=1$