Find the point on the curve y=¼x²+1 which are nearest to the point (0,6). Answer--(±2√3,4)

Hi Shaleen,

Let (x,y) be any point on the curve.

To minimise the distance D = √[(x-6)^2 + y^2]

ie minimise (x-6)^2 + y^2.

Now y = x^2/4 + 1

So substitute for y, and you will get a function in x, say f(x).

For maxima or minima f[dash](x) = 0.

So solve for x from that equation, and you would get two values of x = ±2√3.

for this x value f[double dash](x) > 0, so minima point.

For these x, values of y from y = x^2/4 + 1 = 2.

Hence you get the point on the curve.

Hope that helps.

Best Regards,

Ashwin (IIT Madras).