$\text{The equation of the ellipse whose axes are coincident with the coordinates axes and which touches the straight lines}$

$3x-2y-20=0\text{ and }x+6y-20=0\text{ is }$

detailed solution

Correct option is A

Let the equation of the ellipse bex2a2+y2b2=1We know that the general equation of the tangent to the ellipse isy=mx±a2m2+b2----(i) Since 3x−2y−20=0 or y=32x−10 is tangent to the ellipse, comparing with (i),  m=32 and a2m2+b2=100 or  a2×94+b2=100 or  9a2+4b2=400-----(ii) Similarly, since x+6y−20=0 , i.e., y=−16x+103 is tangent to the ellipse, comparing with (i),  m=16 and a2m2+b2=1009 or  a236+b2=1009 or  a2+36b2=400-----(iii) Solving (ii) and (iii), we get a2=40 and b2=10 .  Therefore, the required equation of the ellipse is x240+y210=1