Mathematics - Parabola Question with Solution | TestHub

: $\mathrm{Let}P\left(t_1,4t_1^2\right),Q\left(t_2,4t_2^2\right)$

Slope of $\mathrm{OP}\times$ slope of $\mathrm{OQ}=-1$

$\Rightarrow 4t_1\cdot 4t_2=-1$

Eq of tangent at $\left(t_1,4t_1^2\right)is$

$y-4t_1^2=8t_1\left(x-t_1\right)\Rightarrow y+4t_1^2=8t_{1}x$

Eq of tangent at $\left(t_2,4t_2^2\right)$ is $y+4t_2^2=8t_{2}x$

Let $\left(x_1,y_1\right)$ is the point of intersection $\begin{array}{rlrl}& eq(1)-eq(2)\Rightarrow x_1=\frac{t_1+t_2}{2}& y_1=8t_1\left(\frac{t_1+t_2}{2}\right)-4t_1^2=4t_1{t}_2=\frac{-1}{4}& \Rightarrow 4y_1+1=0\end{array}$