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
(yq)2=4px
B
(xq)2=4py
C
(yp)2=4qx
D
(xp)2=4qy

Solution

## The correct option is D $${ \left( x-p \right) }^{ 2 }=4qy$$Let the variable circle be$${ x }^{ 2 }+{ y }^{ 2 }+2gx+2fy+c=0$$    ...(1)$$\therefore { p }^{ 2 }+{ q }^{ 2 }+2gp+2fy+c=0$$   ...(2)Circle (1) touches x-axis,$$\therefore { g }^{ 2 }-c=0\Rightarrow c={ g }^{ 2 }$$ From (2) $${ p }^{ 2 }+{ q }^{ 2 }+2gp+2fq+{ g }^{ 2 }=0$$   ...(3)Let the other end of diameter through $$(p,q)$$ be $$(h,k)$$, then$$\displaystyle \frac { h+p }{ 2 } =-g$$ and $$\displaystyle \frac { k+q }{ 2 } =-f$$Put in (3)$$\displaystyle { p }^{ 2 }+{ q }^{ 2 }+2p\left( -\frac { h+p }{ 2 } \right) +2q\left( -\frac { k+q }{ 2 } \right) +{ \left( \frac { h+p }{ 2 } \right) }^{ 2 }=0\\ \Rightarrow { h }^{ 2 }+{ p }^{ 2 }-2hp-4kq=0$$$$\therefore$$ Locus of $$(h,k)$$ is$${ x }^{ 2 }+{ p }^{ 2 }-2xp-4yk=0\Rightarrow { \left( x-p \right) }^{ 2 }=4qy$$Mathematics

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