OCR MEI
Further Pure with Technology
2022
June
— Question 1
Exam Board
OCR MEI
Module
Further Pure with Technology (Further Pure with Technology)
Year
2022
Session
June
Topic
Polar coordinates
1
A family of curves is given by the equation
$$x ^ { 2 } + y ^ { 2 } + 2 a x y = 1 ( * )$$
where the parameter \(a\) is a real number.
You may find it helpful to use a slider (for \(a\) ) to investigate this family of curves.
On the axes in the Printed Answer Booklet, sketch the curve in each of the cases
\(a = 0\)
\(a = 0.5\)
\(a = 2\)
(ii) State a feature of the curve for the cases \(a = 0 , a = 0.5\) that is not a feature of the curve in the case \(a = 2\).
(iii) In the case \(a = 1\), the curve consists of two straight lines. Determine the equations of these lines.
Find an equation of the curve (*) in polar form.
Hence, or otherwise, find, in exact form, the area bounded by the curve, the positive part of the \(x\)-axis and the positive part of the \(y\)-axis, in the case \(a = 2\).
In this part of the question \(m\) is any real number.
Describing all possible cases, determine the pairs of values \(a\) and \(m\) for which the curve with equation (*) intersects the straight line given by \(y = m x\).