KEAM 2011 Math Paper

First we check which point satisfy the equation of hyperbola. All points in options are satisfied the equation of hyperbola 3x2−4y2=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)=‌

11

√13

P(6,3)=‌

25

√13

P(−6,−3)=‌

23

√13

P(6,−3)=‌

13

√13

P(√24,0)=‌

3√24+1

√13

The minimum length is P(−6,3). So, the point (−6,3) is nearest to the given line.