Question

A variable circle passes through the fixed point A (p, q) and touches x-axis. The locus of the other end of the diameter through A is

• A. $(x - p)^2 = 4qy$
• B. $(x - q)^2 = 4py$
• C. $(y - p)^2 = 4qx$
• D. $(y - q)^2 = 4px$

Answer 1

Answer: (A)

$(x - p)^2 = 4qy$

Answer 2

Let the other end of diameter is (h, k) then equation of circle is $(x - h)(x - p) + (y - k)(y - q) = 0$ Put $y = 0$, since x-axis touches the circle $? x^2 - (h + p)x + (hp + kq) = 0 ? (h + p)^2 = 4(hp + kq) \quad (D = 0)$ $? (x - p)^2 = 4qy$.