Question

The largest distance of the point $(a,0)$ from the curve $2x^2+y^2-2x=0$, is given by

• A. $\sqrt{\left(1-2a+a^2\right)}$
• B. $\sqrt{\left(1+2a+2a^2\right)}$
• C. $\sqrt{\left(1+2a-a^2\right)}$
• D. $\sqrt{\left(1-2a+2a^2\right)}$

Answer 1

Answer: (D)

Let $(x,y)$ be the point on the curve $2x^2+y^2-2x=0$. Then its distance from $(a,0)$ is given by
$S=\sqrt{\left\{{\left(x-a\right)}^2+y^2\right\}}...\left(i\right)$
$\Rightarrow S^2=x^2-2ax+a^2+2x-2x^2\left[Using2x^2+y^2-2x=0\right]$
$\Rightarrow S^2=-x^2+2x\left(1-a\right)+a^2\Rightarrow 2S\frac{dS}{dx}=-2x+2\left(1-a\right)$
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 $\frac{d^{2}S}{d{x}^2}<0$ for $x=1-a.$
Hence, $S$ is maximum for $x=1-a$. Putting $x=1-a$ in $\left(i\right).$
We ge $=S=\sqrt{\left(1-2a+2a^2\right)}$