Question

The equation of the chord of contact of the pair of tangents drawn to the ellipse ${4x}^2+{9y}^2=36$ from the point $(m,n)$ where $m.n=m+n$, m , being non positive integer is,

Answer 1

We have,

Equation of ellipse is

$4x^2+9y^2=36$

$\Rightarrow \frac{4x^2+9y^2}{36}=1$

$\Rightarrow \frac{x^2}{9}+\frac{y^2}{4}=1......\left(1\right)$

Given that,

$m(n-1)=n$

So,

$n$ is divisible by $n-1$ .

Then,

$n=2,m=2$

The points are $(2,2)$

Equation of ellipse is

$\frac{x^2}{9}+\frac{y^2}{4}=1$

Now, we know that,

Equation of tangent is

$\frac{x{x}_1}{9}+\frac{y{y}_1}{4}=1$

Then , point $(x_1,y_1)=(2,2)$

Equation of tangent is

$\frac{2x}{9}+\frac{2y}{4}=1$

$\Rightarrow 4x+9y=18$

Hence, this is the answer.