Question

Let $a,s,t$ be nonzero real numbers. Let $P(a{t}^2,2at),\text{and}S(a{s}^2,2as)$ be distinct points on the parabola $y^2=4ax$. If $st=1$, then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

• A. $\frac{(t^2+1{)}^2}{2t^3}$
• B. $\frac{a(t^2+1{)}^2}{2t^3}$
• C. $\frac{a(t^2+1{)}^2}{t^3}$
• D. $\frac{a(t^2+2{)}^2}{t^3}$

Answer 1

The correct option is B $\frac{a(t^2+1{)}^2}{2t^3}$

Normal at $S(a{s}^2,2as)$
$y=a{s}^3+2as-sxy-\frac{a}{t^3}-\frac{2a}{t}=\frac{-x}{t}\cdots \left(1\right)$

Tangent at $P(a{t}^2,2at)$
$ty=x+a{t}^2\Rightarrow y-at=\frac{x}{t}\cdots \left(2\right)$

From equation $\left(1\right)$ and $\left(2\right)$,
$y-at=-y+\frac{a}{t^3}+\frac{2a}{t}\Rightarrow 2y=a(t+\frac{2}{t}+\frac{1}{t^3})\Rightarrow y=\frac{a(t^4+2t^2+1)}{2t^3}=\frac{a(t^2+1{)}^2}{2t^3}$