# Ellipse Geometry

Ellipse Geometry is used for proving ellipse questions relating to geometry, such as similar triangles and circle geometry.

### Worked on Examples of Ellipse Geometry

The diagram shows the ellipse $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a>b$.

(a)    Prove that $SQ = RQ$.

The reflection property of the ellipse indicates that the lines drawn to the points of contact of a tangent from the foci have equal angles of incidence and reflection.
\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \angle \ell PS’ &= \angle SPQ &\color{red} \text{reflection property of the ellipse} \\ \angle \ell PS’ &= \angle RPQ &\color{red} \text{vertically opposite angles} \\ \angle SPQ &= \angle RPQ \\ \angle PQS &= \angle PQR &\color{red} \text{both } 90^{\circ} \\ \triangle SPQ &\equiv \triangle RPQ &\color{red} \text{congruent triangles} \\ \therefore SQ &= RQ \end{aligned}

(b)    Show $S’R = 2a$.

Let $P$ be located at the right-most position at the $x$-axis, and let this point be $V$.
\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} S’R &= S’P + PR \\ &= S’P + SP &\color{red} \triangle SPQ \equiv \triangle RPQ \\ &= S’V + SV &\color{red} \text{definition of ellipse} \\ &= (ae + a) + (a-ae) \\ &= 2a \\ \therefore S’R &= 2a \end{aligned}

(c)    Prove $\triangle OSQ$ and $\triangle S’SR$ are similar.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \angle OSQ &= \angle S’SR &\color{red} \text{common angle} \\ \frac{OS}{S’S} &= \frac{1}{2} &\color{red} OS’ = ae \text{ and } OS = ae \\ \frac{SQ}{SR} &= \frac{1}{2} &\color{red} SQ = QR \\ \therefore \triangle OSQ &\text{ and } \triangle S’SR \text{ are similar }. \end{aligned}

(d)    Prove $Q$ lies on the circle $x^2 + y^2 = a^2$.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \frac{OQ}{S’R} &= \frac{1}{2} \\ OQ &= \frac{1}{2} \times S’R \\ &= \frac{1}{2} \times 2a \\ OQ &= a \\ \text{Therefore } &Q \text{ lies on the circle centre } O \text{ radius }a. \\ \therefore x^2 + y^2 &= a^2 \end{aligned} 