Question

Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

Answer 1

The proof involves setting up the equations for the ellipse, a focus, a directrix, a point on the ellipse, the tangent at that point, the line joining the center to the point of contact, and the perpendicular from the focus to the tangent. Solving the system of equations for the line joining the center to the point of contact and the perpendicular from the focus to the tangent yields an intersection point whose x-coordinate is $a/e$, which is the equation of the corresponding directrix. This confirms that the intersection point lies on the directrix.

Answer 2

Let the ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Focus: $S(ae, 0)$. Directrix: $D: x = \frac{a}{e}$. Point on ellipse: $P(a \cos \theta, b \sin \theta)$.

Tangent at $P$: $\frac{x \cos \theta}{a} + \frac{y \sin \theta}{b} = 1$

Line joining center $O(0,0)$ to $P$: $y = \frac{b \sin \theta}{a \cos \theta} x \implies b x \sin \theta - a y \cos \theta = 0 \quad (*)$

Perpendicular from $S(ae, 0)$ to the tangent: The slope of the tangent is $m_t = -\frac{b \cos \theta}{a \sin \theta}$. The slope of the perpendicular is $m_p = \frac{a \sin \theta}{b \cos \theta}$. Equation of the perpendicular: $y - 0 = \frac{a \sin \theta}{b \cos \theta} (x - ae) \implies a x \sin \theta - b y \cos \theta = a^2 e \sin \theta \quad (**)$

Solving () and (**) for the intersection point: From (), $y = \frac{b \sin \theta}{a \cos \theta} x$. Substitute into (**): $a x \sin \theta - b \left(\frac{b \sin \theta}{a \cos \theta} x\right) \cos \theta = a^2 e \sin \theta$ $a x \sin \theta - \frac{b^2 \sin \theta}{a} x = a^2 e \sin \theta$ Divide by $\sin \theta$ (assuming $\sin \theta \neq 0$): $a x - \frac{b^2}{a} x = a^2 e$ $x \left(\frac{a^2 - b^2}{a}\right) = a^2 e$ Using $a^2 - b^2 = a^2 e^2$: $x \left(\frac{a^2 e^2}{a}\right) = a^2 e \implies x (a e^2) = a^2 e \implies x = \frac{a}{e}$ The x-coordinate of the intersection is $a/e$, which is the equation of the directrix.