A point P moves in such a way that the sum of the slopes of the normals drawn from itto the hyperbola xy = 4 is equal to the sum of the ordinates of feet of the normals. The locus of P is a parabola x 2 = 4y. Then the least distance of this parabola from the circle x 2 – y 2 – 24x + 128 = 0 is -
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The distance of the parabola from the circle means the distance of any point on the parabola from the centre of the circle.
Let A(2t, t 2 ) be any point on the parabola x 2 = 4y and C(12, 0) be the centre of the circle. Then, AC 2 = (2t – 12) 2 + (t 2 – 0) 2 Let Z = AC
2 . Then,
Z = 4(t – 6) 2 + t 4 ∴ 
= 8 (t – 6) + 4t
3 and 
= 8 + 12t 2 For maximum and minimum values of Z, we must have 
⇒ 8(t – 6) + 4t
3 = 0
⇒ t 3 + 2t – 12 = 0
⇒ (t – 2) (t 2 + 2t + 6) = 0
⇒ t = 2
Clearly, 
> 0 for t = 2.
Thus Z is minimum when t = 2. The minimum value of Z is given by Z = 4 (2 – 6) 2 + 2 4 = 80
⇒ AC 2 = 80 ⇒ AC = 4 
Hence, the least distance = 4 
. 
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