Q.
The coordinates of the point on the parabola y2=8x which is at minimum distance from the circle x2+(y+6)2=1 are
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a
(2, −4)
b
(18, −12)
c
(2, 4)
d
None of these
answer is A.
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Detailed Solution
Let P(2t2 , 4t) be any point on the parabola. The centre of the given circle is O (0, -6) and the radius is 1OP2=4t4+(4t+6)2=4t4+4t2+9+12t=4x, where x=t4+4t2+12t+9dxdt=4t3+8t+12=4t3+2t+3=4(t+1)t2−t+3 So dxdt=0⇒t=−1 (other roots are imaginary) So d2xdt2=43t2+2,d2xdt2t=−1>0 Hence OP2 is minimum at t=−1. But if A is any point on the circle and on OP (min), then AP will be minimum when OP is minimum as AP=OP - (radius of circle), Thus the required point is P(2(−1))2,4(−1)=(2,−4)
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