Question

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)

Answer 1

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); (x₁ = 2,y₁ = 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).