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Question

Find the point on the straight line x+y=4, which is nearest to the parabola y2=4(x-9).


A
(8,-4)
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B
(8,4)
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C
(-4,8)
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D
(4,8)
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Solution

The correct option is A (8,-4)



The fig. shows a point A on the parabola where the tangent to the parabola is parallel to the given line x+y=4(1)
It is clear that A is the point on the parabola which is nearest to the given straight line. The slope of the tangent at any point on the parabola y2=4(x9)(2)
Is given by the equation dydx=2y(3)

If (h,k) be the coordinates of A, then 2k=1 [tangent at A is parallel to the given line x+y=4]

i.e. k=2

and putting the value of k in equation (2), we have h=k24+9=10
From the figure we see that if the normal drawn at A intersects the given line at B, then B is the required point on the line nearest to the parabola. Equation of the normal at A is y+2=1(x10) [slope of normal at A=1]
i.e. xy=12(4)
Solving equations (1) and (4) simultaneously gives the coordinates of B as (8,4) which is the required point.


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