Find $a$ such that family of $x^2+ay^2=r^2$ is orthogonal to $y=5x^2$

The derivatives of the ellipse and the parabola are, respectively,

$$y_e' = -\frac x{ay}, \>\>\>\> y_p'=10x$$

Their orthogonality requires $y_w'y_p' = -1$, or

$$ 10x^2 = ay$$

Plug it into $y=5x^2$ to obtain $a = 2$. Thus, the family of $x^2+2y^2 =r^2$ is orthogonal to $y=5x^2$. Shown below is an example for $r=5$.

enter image description here

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Calculus

Derivatives

Ordinary Differential Equations

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