# The point on the curve y^2 = 4x which is nearest to the point (2, 1) is : A. (1,2√2) B. (1, 2) C. (1, –2) D. (–2, 1)

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The point on the curve y2 = 4x which is nearest to the point (2, 1) is :

A. (1,2√2)

B. (1, 2)

C. (1, –2)

D. (–2, 1)

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Option : (B)

If we consider the above figure as for getting it,their will be many points on the curve.

If the normal to the point B passes through the point(2,1) then point B will be the point having nearest distance from point (2,1).

Let B(x,y)

$\frac{dy}{dx}$ = $\frac{2}{y}$

Slope at the point B is (2/y) and normal’s slope will be m=(–y/2) so by point slope formula.

⇒ (y - y1) = m(x - x1); (x= 2,y= 1)

⇒ (y - 1) = (-y/2)(x - 2)

⇒2y - 2 = -xy + 2y

⇒ xy = 2; y2 = 4x

⇒ from above to equations y3 = 8

⇒ y = 2 and x =1

So the nearest point is (1,2).