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

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

Grade:12

1 Answers

Vikas TU
14149 Points
5 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:
ax​2​​+2hxy+by​2​​+c=0
tan2 theta =​a−b​​2h​​ we know so here, tan2 theta =​a−b​​2h​​=​2−10​​6​​=-43​​
Solving tan theta=3 or tan theta=−​3​​1​​.
finding their lengths:
x=rcos theta ,y=rsin theta .ellipse is:
2r​2​​cos​2​​ theta +10r​2​​sin​2​​ theta +6r​2​​sin theta cos theta =1
=>r​2​​(2cos​2​​ theta +10sin​2​​ theta +6sin theta cos theta )=sin​2​​ theta +cos​2​​ theta
and r​2​​=​2+10tan​​​ theta +6tan theta +tan​​​ theta ​​
when tan theta =tan theta​​=3, we get r​​​=​11​​1​​
tan theta =tan theta ​​=−​3​​1​​, we get r​​=​1.
so, the semi-axes are 1 and ​sqrt11​​​​​1​​ .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)

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