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Question

A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle, is :

A
a straight line
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B
an ellipse
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C
a parabola
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D
a hyperbola
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Solution

The correct option is C a parabola
Let the centre of the circle is C(h,k) and radius R.

Now from ΔCAB
k2+(2a)2=R2
k2+4a2=R2 (1)

CD=R
(h0)2+(k2b)2=R2
h2+k2+4b24kb=R2 (2)

Subtract equation (1) from equation (2)
h24kb+4b24a2=0
Replace (h,k) with (x,y)
x24yb+4b24a2=0
(x2+4b2)=4(by+a2)
Hence, it is a parabola.

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