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 ⇒(h−0)2+(k−2b)2=R2 ⇒h2+k2+4b2−4kb=R2⋯(2)
Subtract equation (1) from equation (2) h2−4kb+4b2−4a2=0 Replace (h,k) with (x,y) ⇒x2−4yb+4b2−4a2=0 ⇒(x2+4b2)=4(by+a2) Hence, it is a parabola.