The longest distance of the point (a, 0) from the curve2x2+y2−2x=0 is given by

# The longest distance of the point  from the curve$2{x}^{2}+{y}^{2}-2x=0$ is given by

1. A

$\sqrt{1-2a+{a}^{2}}$

2. B

$\sqrt{1+2a+2{a}^{2}}$

3. C

$\sqrt{1+2a-{a}^{2}}$

4. D

$\sqrt{1-2a+2{a}^{2}}$

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### Solution:

Let  be the point on the curve $2{x}^{2}+{y}^{2}-2x=0$

Then its distance from  is given by

For S to be maximum, we must have

$\frac{dS}{dx}=0⇒-2x+2\left(1-a\right)=0⇒x=1-a$

It can be easily checked that

Hence, Sis maximum for

Putting  in (i), we get $S=\sqrt{1-2a+2{a}^{2}}$

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