find the pint on the curve y^2=4x which is nearest to the point (2, 8)

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2 Answers
Let (x1, y1) be the nearest point on the curve to the given point.
So there are two methods
First one: Find the distance between the two point and minimize it.
Second: The point will be nearest if normal at (x1, y1) passes through given point (Why?)
\hspace{16}$Using Manish Sir hint......\\\\\\First we will Calculate the parametric coordinate of $\bf{y^2=4x}$\\\\\\ which is $\bf{x=t^2}$ and $\bf{y=2t}$\\\\\\ So Coordibate of point $\bf{P(t^2,2t)}$ and Let $\bf{Q(2,8)}$\\\\\\ Now Distance b/w $\bf{PQ = \sqrt{(t^22)^2+(2t+8)^2}}$\\\\\\ Now we have to Minimize $\bf{PQ}$\\\\\\ Now Let $\bf{f(t) = (PQ)^2 = (t^22)^2+4(t+4)^2}$\\\\\\ bcz Minimizing $\bf{PQ}$ means Minimize $\bf{(PQ)^2}$\\\\\\ So Using First Derivative Test.....\\\\\\ $\bf{f^{'}(t) = 2(t^22)\cdot 2t+4\cdot 2 \cdot (t+4)}$\\\\\\
\hspace{16}$For Max. and Min. put $\bf{f^{'}(t)=0\Rightarrow t^32t+2t+8=0}$\\\\\\ we Get $\bf{t^3+2^3=0\Rightarrow t = 2}$ and $\bf{2}$ Complex values.\\\\\\ Now $\bf{f^{''}(t)=......,}$ which is $\bf{>0}$ for $\bf{t=2}$\\\\\\ So $\bf{t=2}$ is a point of Minimum.\\\\\\ So $\bf{f(2)=(PQ)^2 = 20\Rightarrow PQ = 4\sqrt{5}}$\\\\\\ Which is occur at $\bf{(x,y) = (4,4)}$