Two tangents to the ellipse x²/25 + y²/16 = 1 are at right angles to each other…

The tangent at $(x_1, y_1)$ is $\dfrac{xx_1}{a^2} \pm \dfrac{yy_1}{b^2} = 1$. With slope $m$: $c^2 = a^2m^2 \pm b^2$. The director circle is $x^2 + y^2 = a^2 \pm b^2$. For $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ the sign of $h^2 - ab$ names the conic when $\Delta \ne 0$.