2006 Paper 2 Q7

Year: 2006
Paper: 2
Question Number: 7

Course: LFM Pure and Mechanics
Section: Parametric equations

Difficulty: 1600.0 Banger: 1500.0

conics ellipse tangent-normal parametric trigonometric geometric proof curve sketching

Problem

An ellipse has equation $\dfrac{x^2}{a^2} +\dfrac {y^2}{b^2} = 1$. Show that the equation of the tangent at the point $(a\cos\alpha, b\sin\alpha)$ is \[ y=- \frac {b \cot \alpha} a \, x + b\, {\rm cosec\,}\alpha\,. \] The point $A$ has coordinates $(-a,-b)$, where $a$ and $b$ are positive. The point $E$ has coordinates $(-a,0)$ and the point $P$ has coordinates $(a,kb)$, where $0 < k < 1$. The line through $E$ parallel to $AP$ meets the line $y=b$ at the point $Q$. Show that the line $PQ$ is tangent to the above ellipse at the point given by $\tan(\alpha/2)=k$. Determine by means of sketches, or otherwise, whether this result holds also for $k=0$ and $k=1$.

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

Difficulty Comparisons: 0

Banger Rating: 1500.0

Banger Comparisons: 0

Show LaTeX source

Problem source

An ellipse has equation $\dfrac{x^2}{a^2}  +\dfrac {y^2}{b^2} =
1$. Show that 
the equation of the tangent at the point $(a\cos\alpha, b\sin\alpha)$ is
\[
y=- \frac {b  \cot \alpha} a \, x + b\, {\rm cosec\,}\alpha\,.
\]

The point $A$ has coordinates $(-a,-b)$, where $a$ and $b$ are
positive. The point $E$ has coordinates $(-a,0)$
and the point $P$ has coordinates $(a,kb)$, where $0 < k < 1$.
The line through $E$ parallel to  $AP$ meets the line $y=b$ at
the point $Q$. Show that the line $PQ$ is tangent to the above ellipse
at the point given by $\tan(\alpha/2)=k$.
Determine by means of sketches, or otherwise, whether this result
holds also for $k=0$ and $k=1$.

Back to List