A) \[\frac{\pi }{3}\]
B) \[\frac{\pi }{6}\]
C) \[\frac{\pi }{2}\]
D) \[\frac{\pi }{8}\]
Correct Answer: C
Solution :
Equation of circle is \[{{(x-7)}^{2}}+{{(y+1)}^{2}}=25\] \[\therefore \] Centre is \[(7,-1)\]and radius is 5. Let \[y=m\text{ }x\]be the tangent on the circle. \[\therefore \]Length of perpendicular from centre is equal to the radius of circle \[\Rightarrow \] \[\frac{7m+1}{\sqrt{1+{{m}^{2}}}}=\pm \,5\] \[\Rightarrow \] \[49{{m}^{2}}+1+14m=25(1+{{m}^{2}})\] \[\Rightarrow \] \[24{{m}^{2}}+14m-24=0\] \[\Rightarrow \] \[12{{m}^{2}}+7m-12=0\] \[\Rightarrow \] \[12{{m}^{2}}+16m-9m-12=0\] \[\Rightarrow \] \[4m(3m+4)-3(3m+4)=0\] \[\Rightarrow \] \[(3m+4)(4m-3)=0\] \[\Rightarrow \] \[{{m}_{1}}=-\frac{4}{3}\]and \[{{m}_{2}}=\frac{3}{4}\] \[\therefore \] \[{{m}_{1}}{{m}_{2}}=-\frac{4}{3}.\frac{3}{4}\] = - 1 \[\Rightarrow \]Tangent is perpendicular.
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