A) \[{{x}^{2}}+{{y}^{2}}\pm 6x\pm 8y=0\]
B) \[{{x}^{2}}+{{y}^{2}}\pm 8x\pm 6y=0\]
C) \[{{x}^{2}}+{{y}^{2}}\pm 12x\pm 16y=0\]
D) \[{{x}^{2}}+{{y}^{2}}\pm 16x\pm 12y=0\]
Correct Answer: A
Solution :
Let the equation of circle be \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] ...(i) Since it passes through origin \[\therefore \] \[c=0\] Also, length of intercepts on x and y axes are 6 and 8 respectively. \[\therefore \] \[2\sqrt{{{g}^{2}}-c}=6\] \[\Rightarrow \] \[\sqrt{{{g}^{2}}-c}=3\] \[\Rightarrow \] \[g=\pm 3\] and \[2\sqrt{{{f}^{2}}-c}=8\] \[\Rightarrow \] \[\sqrt{{{f}^{2}}}=4\] \[\Rightarrow \] \[f=\pm 4\] From Eq. (i) \[{{x}^{2}}+{{y}^{2}}\pm 2(3)x\pm 2(4)y+0=0\] \[\Rightarrow \] \[{{x}^{2}}+{{y}^{2}}\pm 6x\pm 8y=0\] which is the required equation of circle.
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