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Q.
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
Let $A\left(x_{1}, x_{1}-5\right)$ be a point on $x-y=5$, then the chord of contact of $x^{2}+4 y^{2}=4$ with respect to $\bar{A}$ is
$x \cdot x_{1}+4 y\left(x_{1}-5\right)=4$
$\Rightarrow \,(x+4 y) x_{1}-(20 y+4)=0$
Since, it passes through a fixed point.
$\therefore \, x+4 y=0$
and $20 y+4=0$
(from $P+\lambda Q=0)$
$\Rightarrow \, y=-\frac{1}{5}$ and $x=\frac{4}{5}$
So, the coordinates of fixed point is $\left(\frac{4}{5},-\frac{1}{5}\right)$.