What is the minimum distance between the circle x^2 + y^2 =9 and curve 2x^2 + 10y^2 +6xy=1 ?

Question

Vikas TU · Answer

Dear Student,

the given curve 2x^2+10y^2+6xy=1 is an ellipse with centre=(0,0)

similarly, curve x^2+y^2=9 is circle with radius =3 and origin as center

finding semi-axes of ellipse:

ax2+2hxy+by2+c=0

tan2 theta =a−b2h we know so here, tan2 theta =a−b2h=2−106=-43

Solving tan theta=3 or tan theta=−31.

finding their lengths:

x=rcos theta ,y=rsin theta .ellipse is:

2r2cos2 theta +10r2sin2 theta +6r2sin theta cos theta =1

=>r2(2cos2 theta +10sin2 theta +6sin theta cos theta )=sin2 theta +cos2 theta

and r2=2+10tan theta +6tan theta +tan theta

when tan theta =tan theta=3, we get r=111

tan theta =tan theta =−31, we get r=1.

so, the semi-axes are 1 and sqrt111 .semi-major axes is 1.

min distance=3-1=2

so the minimum distance between the 2 curves is 2 units [Ans].

Cheers!!

Regards,

Vikas (B. Tech. 4^th year
Thapar University)

1 Answers