The perpendicular distance from the origin to the focal chord drawn through the point $(4,5)$ to the…
The perpendicular distance from the origin to the focal chord drawn through the point $(4,5)$ to the parabola $y^2-4 y-3 x+7=0$ is
$\frac{2}{5}$
$\frac{1}{\sqrt{2}}$
$\frac{1}{5}$
$1$
Solution
Given equation of parabola
$\begin{aligned}
& y^2-4 y-3 x+7=0 \Rightarrow(y-2)^2=3(x-1) \\
& \Rightarrow(y-2)^2=4 \cdot \frac{3}{4}(x-1)
\end{aligned}$
So focus is $\left(\frac{3}{4}+1,2\right)=\left(\frac{7}{4}, 2\right)$
Now, equation of focal chord is
$y-5=\frac{2-5}{\frac{7}{4}-4}(x-4) \Rightarrow 4 x-3 y-1=$
Distance between focal chord \& origin is
$\frac{|0-0-1|}{\sqrt{(-3)^2-4^2}}=\frac{1}{5}$