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Question

A variable tangent to the parabola y2=4ax meets the circle x2+y2=r2 at P & Q .Then the locus of the mid point of PQ is x(x2+y2)+ay2.

This statement is ___.

A
True
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B
False
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Solution

The correct option is A True
Equation of tangent to parabola y2=4ax at point (x1,y1) is given by yy1=2a(x+x1) ---(1)
Also (x1,y1) lies on parabola
y21=4ax1 (2)
Equation of given circle is
$x^2+y^2=r^2.$
Let (h,k) be the mid point of PQ
equation of chord whose mid point is (h,k) is given by T=x1 i.e xh+yk=h2+k2 (3)
equation equation (1) and (3) are identical.
Therefore 2ah=y1k=2ax(h2+k2)
y1=2akh (4)
and x1=(h2+k2)h (5)
putting the value of x1 and y1 from equation (4) and (5) is (2).
we get, (2akh)2=4a((h2+k2)h)
4a2k2h2=4ah(h2+k2)
4ah(h2+k2)+4a2k2=0
ah(h2+k2)+a2k2=0
Locus of mid-point of PQ is given by x(x2+y2)+ay2=0
This statement is True.

2117459_1131484_ans_6734be02d433461686ea126dc4fe0346.png

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