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Q.
The largest distance of the point $(a, 0)$ from the curve $2x^2 + y^2 - 2x = 0$, is given by
Application of Derivatives
Solution:
Let $(x, y)$ be the point on the curve $2x^2 + y^2 -2x = 0$. Then its distance from $(a, 0)$ is given by
$S=\sqrt{\left\{\left(x-a\right)^{2}+y^{2}\right\}}\,...\left(i\right)$
$\Rightarrow S^{2} = x^{2} - 2ax + a^{2} + 2x - 2x^{2} \left[Using\, 2x^{2} + y^{2} - 2x = 0\right]$
$\Rightarrow S^{2}=-x^{2}+2x\left(1-a\right)+a^{2} \Rightarrow 2S \frac{dS}{dx}=-2x+2\left(1-a\right)$
For S to be maximum,
$\frac{dS}{dx}=0 \Rightarrow -2x+2\left(1-a\right)=0 \Rightarrow x=1-a$
It can easily checked that $\frac{d^{2}S}{dx^{2}} < 0$ for $x = 1 - a.$
Hence, $S$ is maximum for $x = 1 - a$. Putting $x = 1- a$ in $\left(i\right).$
We ge $=S=\sqrt{\left(1-2a+2a^{2}\right)}$