Question

Find the point on the curve y^2=4x which is nearest to the point (2,1)

Answer 1

(1,2) is the nearest point to it.

Answer 2

• For a point to be nearest the distance between the point on the curve and point(2,1) should be minimum.
• Distance = √((x-2)²+(y-1)²)
• Now x=y²/4

So new equation of distance=D= √(((y²/4)-2)²+(y-1)²)

• For minimum distance  $\frac{dD}{dy}$=0

⇒$\frac{1}{2}$(((y²/4)-2)²+(y-1)²)^-.5(2((y²/4)-2)y/2+2(y-1))=0

⇒y³/4-2y+2y-2=0

⇒y³=8

⇒y=2

⇒x=y²/4

    x=2²/4

    x=1

∴ Point on the curve nearest to the point (2,1) is (1,2).