The locus of the intersection point of two perpendicular tangents to a given ellipse

It is a well known circle, generally called the "Director circle", though Wikipedia prefers "Fermat–Apollonius circle"

You can find a projective geometric proof here from Paris Pamfilos though I prefer one based on analytical geometry such as the discussion here by Michael Raugh or the following from slides 12 and 13 from Career Launcher India using earlier results for tangents to an ellipse:

If the ellipse is $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ and we take a tangent to the ellipse from the point $(h,k)$ with slope $m$ then $(h^2-a^2)m^2-2hkm+(k^2-b^2) =0$.

This has two solutions $m_1$ and $m_2$ with $m_1 m_2 = \dfrac{k^2-b^2}{h^2-a^2}$, and the two tangents are perpendicular if $m_1 m_2 = -1$ so we need $h^2 +k^2 = a^2+b^2$, meaning the locus is the circle $$x^2+y^2 = a^2+b^2.$$

If all you want is a proof that the locus is a circle, we may assume that the ellipse is given by $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$

Ignoring vertical tangents for now, if a line $y=mx+k$ is tangent to the ellipse, then plugging in this value of $y$ into the equation for the ellipse gives $$\frac{x^2}{a^2} + \frac{(m^2x^2 + 2mkx + k^2)}{b^2} = 1$$ or $$(b^2 + a^2m^2)x^2 + 2a^2mkx + (a^2k^2 - a^2b^2) = 0.$$ This equation gives the two points of intersection of the line with the ellipse. If the line is tangent, then the two points must coincide, so the quadratic must have zero discriminant. That is, we need $$(2a^2mk)^2 - 4(a^2k^2 - a^2b^2)(b^2+a^2m^2) = 0$$ or equivalently, $$\begin{align*} (a^2m^2)k^2 -a^2(b^2+a^2m^2)k^2 &= -a^2b^2(b^2+a^2m^2)\\ -a^2b^2k^2&= -a^2b^2(b^2+a^2m^2)\\ k^2 &= b^2+a^2m^2\\ k &= \pm\sqrt{a^2m^2 + b^2}. \end{align*} $$ So the lines that are tangent to the ellipse are of the form $$y = mx \pm \sqrt{a^2m^2 + b^2}.$$

Since the problem is symmetric about $x$ and $y$, consider the points on the upper half plane, so that we will take the plus sign above. The tangent perpendicular to this one will therefore have equation $$y = -\frac{1}{m}x + \sqrt{\frac{a^2}{m^2} + b^2},$$ or equivalently $$my = -x + \sqrt{a^2 + m^2b^2}.$$ (We are ignoring the vertical and horizontal tangents; I'll deal with them at the end).

If a point $(r,s)$ is on both lines, then we have $$\begin{align*} s-mr &= \sqrt{a^2m^2 + b^2}\\ ms + r &= \sqrt{a^2+m^2b^2}. \end{align*}$$ Squaring both sides of both equations we get $$\begin{align*} s^2 - 2mrs + m^2r^2 &= a^2m^2 + b^2\\ m^2s^2 + 2mrs + r^2 &= a^2 + m^2b^2 \end{align*}$$ and adding both equations, we have $$\begin{align*} (1+m^2)s^2 + (1+m^2)r^2 &= (1+m^2)a^2 + (1+m^2)b^2,\\ (1+m^2)(s^2+r^2) &= (1+m^2)(a^2+b^2)\\ s^2+r^2 = a^2+b^2, \end{align*}$$ showing that $(s,r)$ lies in a circle, namely $x^2+y^2 = a^2+b^2$.

Taking the negative sign for the square root leads to the same equation.

Finally, for the vertical and horizontal tangents, these occur at $x=\pm a$; the horizontal tangents are $y=\pm b$. Their intersections occur at $(\pm a,\pm b)$, which lie on the circle given above. So the locus of such points is contained in the circle $x^2+y^2 = a^2+b^2$.

Conversely, consider a point $(r,s)$ that lies on $x^2+y^2 = a^2+b^2$. If a tangent to the ellipse $$ y = mx + \sqrt{a^2m^2 + b^2}$$ goes through $(r,s)$, then we have $$ s-mr = \sqrt{a^2m^2+b^2}.$$ Squaring both sides, we have $$s^2 - 2msr + m^2r^2 = a^2m^2 + b^2$$ or $$(a^2-r^2)m^2 +2srm + (b^2-s^2) = 0.$$ Since $r^s+s^2 = a^2+b^2$, then $a^2 - r^2 = s^2-b^2$, we we have $$(s^2-b^2)m^2 + 2srm + (b^2-s^2) = 0,$$ and if we do not have $s=\pm b$ (the horizontal/vertical tangent intersection points), then we get $$m^2 + tm - 1 = 0,\qquad\text{where } t = \frac{2sr}{s^2-b^2}.$$ So the two solutions for $m$, $m_1$ and $m_2$, satisfy $m_1m_2 = -1$, hence the two tangents are perpendicular. That is, at every point on the (upper half of the) circle, the two lines through the point that are tangent to the ellipse are perpendicular to each other.

So all such points are on the circle, and all points on the circle are such points. (The circle is called the director circle of the ellipse).

Not a synthetic proof, but ...

It's interesting to note that four target points are easy to identify: these are the corners of the rectangle that bounds the ellipse on the left, right, top, and bottom; call this the "axis-aligned bounding rectangle". Clearly, if we rotate the ellipse about its center, then the corners of the (new, un-rotated) axis-aligned bounding rectangle of the rotated figure also correspond to four target points. We can show that the target points form a circle by showing that these axis-aligned rectangles always have semi-diagonal $\sqrt{a^2 + b^2}$, regardless of rotation.

Parameterizing the ellipse by $P(a\cos\theta,b\sin\theta)$ and rotating about the center by angle $\phi$ gives an ellipse parameterized by

$$(a \cos\theta \cos\phi - b \sin\theta \sin\phi, a \cos\theta \sin\phi + b \sin\theta \cos\phi )$$

The right side of the axis-aligned bounding rectangle is determined by the value of $\theta$ that maximizes the $x$ coordinate; the top side is determined by the value that maximizes the $y$ coordinate.

Note that both parameterized coordinates have the form $U\cos\theta + V\sin\theta$, which can be expressed as $\sqrt{U^2+V^2} \sin\left(\theta+\psi\right)$ for some $\psi$. As $\theta$ traverses all values from $0$ to $2\pi$, the coordinate formula must attain a maximum value of $\sqrt{U^2 + V^2}$ (and we needn't worry about the exact value of $\psi$).

Consequently, the rotated ellipse's maximal $x$-coordinate is $x_{\star}:=\sqrt{a^2\cos^2\phi + b^2\sin^2\phi}$ and its maximal $y$-coordinate is $y_{\star}:=\sqrt{a^2\sin^2\phi + b^2\cos^2\phi}$, so that the rectangle's semi-diagonal, $d$, satisfies

$$d^2 = x_{\star}^2 + y_{\star}^2 = \left( a^2 + b^2 \right)\left( \cos^2\phi + \sin^2\phi \right) = a^2 + b^2$$

Therefore, the corners of the axis-aligned bounding rectangles of the rotated ellipse lie on the circle of radius $\sqrt{a^2 + b^2}$, regardless of the angle of rotation, $\phi$; rotating them back through angle $\phi$ won't change that fact.

This shows that all of the points with the "perpendicular tangents" property lie on a particular circle, but it does not show that all points on that circle have the "perpendicular tangents" property. That implication is left to the reader.