Question

Tangent are drawn from the points on the line $x-y-5=0$ to $x^2+4y^2=4$, then all the chords of contact pass through a fixed point, whose coordinate are

• A. $(\frac{4}{5},-\frac{1}{5})$
• B. $(\frac{4}{5},\frac{1}{5})$
• C. $(-\frac{4}{5},\frac{1}{5})$
• D. None of the these

Answer 1

The correct option is C $(-\frac{4}{5},\frac{1}{5})$

Equation of ellipse is given $x^2+4y^2=4$
Equation of chord of contact is $T=0$
$x{x}_1+4y{y}_1-4=\mathrm{0...}\left(1\right)$ which intersect with line $x-y-5=0$
Let $x=k\Rightarrow y=k-5$
From equation $\left(1\right)$
$xk+4y(k-5)-4=0$
$\Rightarrow k(x+4y)-4(5y+1)=0$
So, $x+4y=0$ and $5y+1=0$
$\Rightarrow y=-\frac{1}{5}$ and $x=-\frac{4}{5}$