A) \[{{(y-q)}^{2}}=4px\]
B) \[{{(x-q)}^{2}}=4py\]
C) \[{{(y-p)}^{2}}=4qx\]
D) \[{{(x-p)}^{2}}=4qy\]
Correct Answer: D
Solution :
Let the variable circle be \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] ....(1) Since this circle is passing through \[A(p,q)\] \[\therefore \,{{p}^{2}}+{{q}^{2}}+2gp+2fq+c=0\] ....(2) Circle (1) touches x-axis, \[\therefore \,\,\,{{g}^{2}}-c=0\Rightarrow c={{g}^{2}}.\] From (2), we have \[{{p}^{2}}+{{q}^{2}}+2gp+2fq+{{g}^{2}}=0\] ....(3) Let the other end of diameter through \[(p,q)\] be \[(h,k),\]then \[\frac{h+p}{2}=-g\] and \[\frac{k+q}{2}=-f\] Put in (3) \[{{p}^{2}}+{{q}^{2}}+2p\left( -\frac{h+p}{2} \right)+2q\left( -\frac{k+q}{2} \right)+{{\left( \frac{h+p}{2} \right)}^{2}}=0\]\[\Rightarrow \,\,{{h}^{2}}+{{p}^{2}}-2hp-4kq=0\] \[\therefore \] locus of \[(h,k)\] is \[{{x}^{2}}+{{p}^{2}}-2xp-4yq=0\] \[\Rightarrow \,\,{{(x-p)}^{2}}=4qy\]
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