Q. 30 4.3( 3 Votes )

Find the point on the curve y2 = 4x which is nearest to the point (2, –8).

Answer :

Given Curve is y2 = 4x …… (1)

Let us assume the point on the curve which is nearest to the point (2, - 8) be (x, y)

The (x, y) satisfies the relation(1)

Let us find the distance(S) between the points (x, y) and (2, - 8)

We know that distance between two points (x1,y1) and (x2,y2) is .

Squaring on both sides we get,

S2 = x2 + y2 - 4x + 16y + 68

From (1)

We know that distance is an positive number so, for a minimum distance S, S2 will also be minimum

Let us S2 as the function of y.

For maxima and minima,

y3 - 64 = 0

On solving we get

Now differentiating again

At y = 4

We get minimum distance at y = 4

Let find the value of x at these y values

x = 4

The nearest point to the point (2, - 8) on the curve is (4,4).

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