Pearson Edexcel IAL Mathematics FP3.2.1 Cartesian & parametric equations

Practise moving between Cartesian and parametric forms of ellipses and hyperbolas to find intersections, midpoints and asymptotes.

Exam points

substitute a line into an ellipse or hyperbola to form a quadratic in x
use root sums to find midpoint coordinates without solving both intersection points
read asymptotes from a hyperbola equation or its parametric form

[Maximum number: 5]

The ellipse E has equation

\frac{x^2}{9}+\frac{y^2}{4}=1

The line l has equation y=kx-3, where k is a constant.
Given that E and l meet at 2 distinct points P and Q,

(a)

show that the x coordinates of P and Q are solutions of the equation

(9k^2+4)x^2-54kx+45=0

The point M is the midpoint of PQ.

[ 2 ]

\frac{x^2}{9}+\frac{(kx-3)^2}{4}=1

4x^2+9(k^2x^2-6kx+9)=36

(9k^2+4)x^2-54kx+45=0

M1 A1*

(b)

Determine, in simplest form in terms of k, the coordinates of M.

[ 3 ]

Using the sum of roots,

x_M=\frac12\left(\frac{54k}{9k^2+4}\right)=\frac{27k}{9k^2+4}

Then

y_M=k\left(\frac{27k}{9k^2+4}\right)-3=\frac{27k^2-27k^2-12}{9k^2+4}=-\frac{12}{9k^2+4}

M\left(\frac{27k}{9k^2+4},-\frac{12}{9k^2+4}\right)

M1 dM1 A1

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