A-LevelFurther MathematicsPolar coordinates9231/12 · Oct/Nov 2022 · Q512 Marks

The curve C has polar equation `r = a sec²θ`, where a is a positive constant and `0 ≤ θ ≤ 1/4π`.

(a)Sketch C, stating the polar coordinates of the point of intersection of C with the initial line and also with the half-line `θ = 1/4π`.[3]
(b)Find the maximum distance of a point of C from the initial line.[2]
(c)Find the area of the region enclosed by C, the initial line and the half-line `θ = 1/4π`.[4]
(d)Find, in the form `y = f(x)`, the Cartesian equation of C.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[B1] Correct position and domain and `r` strictly increasing.
[B1] Decreasing positive gradient.
[B1] `(a,0)` `(2a, 1/4π)` (Can be labelled on their sketch.)
[M1] Uses `y = r sin θ`
[A1] `a√2`
[M1] Forms `1/2∫ r² dθ` with correct limits.
[M1] Applies `sec²θ = tan²θ+1` to obtain an integrable form.
[A1] `1/2a² [(tan³θ)/3 + tanθ]₀^(1/4π)`
[A1] `2/3a²`
[M1] Substitutes to eliminate either `r` or `θ` (`x = r cosθ`, `y = r sinθ`, `r = a sec²θ` -> `x² = ar`).
[M1] Completes substitution to eliminate both `r` and `θ` and makes `y` or `y²` the subject.
[A1] `y = x/a√(x²-a²)` or `y² = (x²)/(a²)(x²-a²)` (AEF)

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