What is the minimum distance between the circle x^2 + y^2 =9 and curve 2x^2 + 10y^2 +6xy=1 ?
Pushkar Aggarwal , 8 Years ago
Grade 12
Analytical Geometry> What is the minimum distance between the ...
1 AnswersVikas TU
Last Activity: 7 Years ago
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. 4th year
Thapar University)
Thapar University)
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