Parabola 2 Question 17
17. If $s t=1$, then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is
(a) $\frac{\left(t^{2}+1\right)^{2}}{2 t^{3}}$
(b) $\frac{a\left(t^{2}+1\right)^{2}}{2 t^{3}}$
(c) $\frac{a\left(t^{2}+1\right)^{2}}{t^{3}}$
(d) $\frac{a\left(t^{2}+2\right)^{2}}{t^{3}}$
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Solution:
- PLAN Equation of tangent and normal at $\left(a t^{2}, 2 a t\right)$ are given by $t y=x+a t^{2}$ and $y+t x=2 a t+a t^{3}$, respectively.
Tangent at $\quad P: t y=x+a t^{2}$ or $y=\frac{x}{t}+a t$
Normal at $S: y+\frac{x}{t}=\frac{2 a}{t}+\frac{a}{t^{3}}$
Solving, $\quad 2 y=a t+\frac{2 a}{t}+\frac{a}{t^{3}} \Rightarrow y=\frac{a\left(t^{2}+1\right)^{2}}{2 t^{3}}$
