circlesconic sectionscoordinate systems
A circle passes through the focii of an ellipse.
Circle intersects ellipse at points A and B. Tangents and normals are drawn to the ellipse at A & B such that tangents intersect at P and normals intersect at Q. Then what is PQ is equal to ?
Is this a property that when a circle passes through focii of ellipse, the tangents meet at P and normals meet at Q and PQ is one of the diameters of the circle?
We may suppose that $$\text{the ellipse$\ :\ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\quad a\gt b\gt 0$}$$ $$\text{the circle$\ :\ x^2+y^2=a^2-b^2$}$$
As you wrote, we have $$2a=17\quad\Rightarrow \quad a=\frac{17}{2}$$
Since $$\frac{a^2-b^2-y^2}{a^2}+\frac{y^2}{b^2}=1\quad\Rightarrow\quad |y|=\frac{b^2}{\sqrt{a^2-b^2}}$$
we have $$30=\frac 12\times 2\sqrt{a^2-b^2}\times \frac{b^2}{\sqrt{a^2-b^2}}\quad\Rightarrow\quad b=\sqrt{30}.$$
Thus, the answer is $$2\sqrt{a^2-b^2}=2\sqrt{\left(\frac{17}{2}\right)^2-30}=\color{red}{13}.$$
This is a property of general ellipses, so let's consider the ellipse with major and minor radii $a$ and $b$. For a given point $P = (a \cos 2\theta, b \sin 2\theta)$, we'll identify the point $Q$ common to all chords $\overline{AB}$ such that $\overline{AP}\perp\overline{BP}$.
It's straightforward to find the coordinates of $Q$ at the intersection of two convenient chords, namely: $\overline{A_0 B_0}$, the "other" diagonal of the inscribed rectangle with vertex $P$; and $\overline{PN}$, along the normal to the ellipse at $P$ (which serves as the degenerate case). $$\begin{align} \overline{A_0 B_0}:&\quad b x \sin 2\theta + a y \cos 2\theta = 0 \\ \overline{PN}:&\quad a x \sin 2\theta - b y \cos 2\theta = ( a^2 - b^2 ) \cos 2\theta \sin 2\theta \end{align}$$ $$Q := \overline{A_0 B_0} \cap \overline{PN} = \frac{a^2-b^2}{a^2+b^2} \;\left( a \cos 2\theta, - b \sin 2\theta \right)$$
Now, consider a generic chord $\overline{AB}$ where $A = (a \cos2\alpha, b \sin 2\alpha)$ and $B = (a \cos2\beta, b \sin 2\beta)$. The condition that $\overline{AP}\perp\overline{BP}$ gives rise to this equation $$\begin{align} 0 &= (A-P)\cdot(B-P) \\ &= 4 \sin(\alpha-\theta) \sin(\beta-\theta) \left(\; a^2 \sin(\alpha + \theta) \sin( \beta + \theta ) + b^2 \cos(\alpha + \theta) \cos(\beta + \theta)\;\right) \end{align}$$ where we may safely ignore the initial factors, so that
$$a^2 \sin(\alpha + \theta) \sin( \beta + \theta ) + b^2 \cos(\alpha + \theta) \cos(\beta + \theta) = 0 \tag{1}$$
On the other hand, the equation for the line containing $\overline{AB}$ is $$b x \cos(\alpha + \beta) + a y \sin(\alpha + \beta) = a b \cos(\alpha - \beta) $$ so that the condition that $Q$ lies on the chord becomes $$\frac{a^2-b^2}{a^2+b^2}\;\left(\;a b \cos(\alpha + \beta) \cos 2\theta - a b \sin(\alpha + \beta) \sin 2\theta \; \right) = a b \cos(\alpha - \beta) $$ whereupon
$$\left(a^2-b^2\right)\cos(\alpha + \beta + 2\theta) = \left(a^2+b^2\right) \cos(\alpha - \beta) \tag{2}$$
Verification that (1) and (2) are equivalent is left as an easy exercise for the reader. $\square$
Best Answer
Consider the ellipse with foci $F_1$ and $F_2$ as shown below, with a circle through $F_1$ and $F_2$ intersecting the ellipse at $A$. Furthermore we have the tangent to the ellipse at $A$, intersecting the circle at $S$, and the perpendicular to the tangent at $A$, intersecting the circle at $R$.
Knowing that $\angle F_1AR = \angle \angle F_2AR,$ what can you conclude about circular arcs $F_1R$ and $F_2R$?
Knowing that $\angle RAS$ is a right angle, what can you conclude about the line segment $RS$?
It can be shown that $R$ and $S$ are the points where the minor axis of the ellipse intersects the circle. Therefore if we repeat this construction at another intersection point of the ellipse and circle, for example $B$ in this figure, the tangent and perpendicular lines will again intersect the circle at two points on the minor axis of the ellipse, which therefore are the same points $R$ and $S,$ which therefore are the two points $P$ and $Q.$
In this case the construction works equally well at any of the four intersection points between the circle and ellipse. If we alter the shape or positions of the ellipse and/or circle so that there are only two intersections, the construction is still valid for those two points.