Another implicit function problem

[Yankel]

Hello, I have one more (hopefully last) question regarding implicit functions:

The ellipse

\[9x^{2}+y^{2}=36\]

and the hyperbole

\[xy=a\]

tangent at a point in the first quarter.

I need to find the tangent point and a.

thanks !

I know that for the ellipse:

\[\frac{\partial y}{\partial x}=-9\frac{x}{y}\]

and for the hyperbole:

\[\frac{\partial y}{\partial x}=-\frac{y}{x}\]

 

[Yankel]

Solved it (Nod)

thanks anyway

 

[MarkFL]

A precalculus technique would be to solve the hyperbola for $y$ and then substitute into the ellipse to get in standard form:

\(\displaystyle 9x^4-36x^2+a^2=0\)

For the two curves to be tangent, we require the discriminant to be zero:

\(\displaystyle 36^2-4\cdot9\cdot a^2=0\)

\(\displaystyle a^2=36\)

Since a tangent point is to be in the first quadrant, we take the positive root:

\(\displaystyle a=6\)

Hence, we have:

\(\displaystyle 9x^4-36x^2+36=0\)

\(\displaystyle x^4-4x^2+4=0\)

\(\displaystyle \left(x^2-2\right)^2=0\)

Taking the positive root:

\(\displaystyle x=\sqrt{2}\implies y=\frac{6}{\sqrt{2}}=3\sqrt{2}\)

And so the point of tangency is:

\(\displaystyle \left(\sqrt{2},3\sqrt{2}\right)\)

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