Let P be a point on the parabola, x2=4y. If the distance of P from the centre of the circle, x2+y2+6x+8=0 is minimum, then the equation of the tangent to the parabola at P, is :
A
x−y+3=0
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B
x+4y−2=0
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C
x+y+1=0
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D
x+2y=0
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Solution
The correct option is Cx+y+1=0 LetP(2t,t2)Centre of the circle : (−3,0)y=D2=(2t+3)2+(t2−0)2For minimum distance, dydt=0⇒2(2t+3)2+4t3=0⇒t3+2t+3=0⇒(t+1)(t2−t+3)=0⇒t=−1∴P is (−2,1)Equation of tangent to the parabola is, xx1=2a(y+y1)⇒x(−2)=2×1(y+1)⇒x+y+1=0