Question

Equation of auxiliary circle of ellipse 2x^2+6xy+5y^2=1

Answer 1

equation 2x² + 6xy + 5y² = 1 is ellipse.
Let an auxiliary circle of given ellipse , x² + y² = a²

so, x = acos$\theta$ , y = asin$\theta$

now, 2a²cos²$\theta$ + 6a²sin$\theta$cos$\theta$ + 5sin²$\theta$ = 1

a²(2cos²$\theta$ + 6sin$\theta$.cos$\theta$ + 5 - 5cos²$\theta$) = 1

a²(5 - 3cos²$\theta$ + 3sin2$\theta$ ) = 1

a²(5 - 3(1 + cos2$\theta$)/2 + 3sin2$\theta$) = 1

a²(10 - 3 - 3cos2$\theta$ + 6sin2$\theta$) = 2

a²(7 + 6sin2$\theta$ - 3cos2$\theta$ ) = 2

here , 7-√45 ≤ 7 + 6sin2$\theta$ - 3cos2$\theta$ ≤ 7 + √45

so, 1/(7 -√45) ≤ 1/(6sin2$\theta$ - 3cos2$\theta$) ≤ 1/(7 + 45) ,

or, 2/(7 - √45) ≤ a² ≤ 2/(7 + √45)

so, equation of auxiliary circle is x² + y² = a² where a² $\in$ [2/(7 - √45), 2/(7 + √45)]