Let $\alpha_1$ and $\alpha_2$ be the ordinates of two points $A$ and $B$ on a parabola $y^2=4 a x$ and let…
Let $\alpha_1$ and $\alpha_2$ be the ordinates of two points $A$ and $B$ on a parabola $y^2=4 a x$ and let $\alpha_3$ be the ordinate of the point of intersection of its tangents at $A$ and $B$. Then, $\alpha_3-\alpha_2=$
$\alpha_3-\alpha_1$
$\alpha_3+\alpha_1$
$\alpha_1$
$\alpha_1-\alpha_3$
Solution
Ordinate of point of intersection of tangents at $A$ and $B$ whose ordinates are $\alpha_1$ and $\alpha_2$ is $\frac{\alpha_1+\alpha_2}{2}$,
So,
$
\begin{aligned}
\alpha_3 & =\frac{\alpha_1+\alpha_2}{2} \\
2 \alpha_3 & =\alpha_1+\alpha_2 \Rightarrow \alpha_3-\alpha_2=\alpha_1-\alpha_3 .
\end{aligned}
$