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 \(x^2 + y^2 = a^2 \).

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


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