A circle cuts the chord on x-axis of length `4a`. If this circle cuts the y-axis at a point whose distance from origin is `2b`. Locus of its centre is

Options

A. Ellipse
B. Parabola
C. Hyperbola
D. Straight line

Answer

A

Explanation

According to given information, we have the following figure.

Let the equation of circle be
`x^(2) + y^(2) + 2 gx + 2 fy + c = 0`
According the problem,
`4a = 2 sqrt(g^(2) - c)`
[`:'` The length of intercepts made by the circle `x^(2) + y^(2) + 2 gx + 2 fy + c = 0`
With X-axis is `2 sqrt(g^(2) - c)`
Also, as the circle is passing through P (0, 2b)
`:. 0 + 4b^(2) + 0 + 4 bf + c = 0`
`implies 4b^(2) + 4bf + c = 0`
Elimainating 'c' from Eqs. (ii) and (iii) we get
`4b^(2) + 4bf + g^(2) - 4a^(2) = 0`
`[:' 4a = 2 sqrt(g^(2) - c) implies = g^(2) - 4 a^(2)]`
so, locus of (-g - f) is
`4b^(2) - 4by + x^(2) - 4a^(2) = 0`
`implies x^(2) = 4by + 4a^(2) - 4b^(2)`
Which is a parabola.