A) \[(-\text{ }6,3)\]
B) (6, 3)
C) \[(-\text{ }6,-3)\]
D) \[(\text{ }6,-3)\]
E) \[(\sqrt{24},0)\]
Correct Answer: A
Solution :
First we check which point satisfy the equation of hyperbola. All points in options are satisfied the equation of hyperbola\[3{{x}^{2}}-4{{y}^{2}}=72\]. Now, we find one-by-one the length of perpendicular from point on Ellipse to the line \[3x+2y+1=0\]. \[{{p}_{(-6,3)}}=\frac{11}{\sqrt{13}}\] \[{{p}_{(6,3)}}=\frac{25}{\sqrt{13}}\] \[{{p}_{(-6,-3)}}=\frac{23}{\sqrt{13}}\] \[{{p}_{(6,-3)}}=\frac{13}{\sqrt{13}}\] \[{{p}_{(\sqrt{24},0)}}=\frac{3\sqrt{24}+1}{\sqrt{13}}\] The minimum length is\[{{p}_{(-6,3)}}\]. So, the point \[(-6,3)\] is nearest to the given line.
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