question_answer

The sum of the minimum distance and the maximum distance from the point\[(4,-3)\]to the circle\[{{x}^{2}}+{{y}^{2}}+4x-10y-7=0\]is

A)  20                                         

B)  12

C)  10               

D)                         16

E)  22

Correct Answer: A

Solution :

Given equation of circle \[{{x}^{2}}+{{y}^{2}}+4x-10y-7=0\] with centre\[=(-2,5)=C\]and radius\[=\sqrt{4+25+7}=\sqrt{36}=6\] Now,     \[PC=\sqrt{{{(4+2)}^{2}}+{{(-3-5)}^{2}}}\] \[=\sqrt{36+64}=\sqrt{100}=10\] So, the point lies outside the circle. Now, maximum distance \[=PQ=PC+QC=10+6=16\] and minimum distance \[=PS=PC-SC=10-6=4\] \[\therefore \]Sum of maximum and minimum distance \[=16+4=20\]