FP1 June 2013 Q6
6. A parabola \(C\) has equation \(y^2 = 4ax,\quad a > 0\)
The points \(P(ap^2,\ 2ap)\) and \(Q(aq^2,\ 2aq)\) lie on \(C\), where \(p \neq 0\), \(q \neq 0\), \(p \neq q\).
(a) Show that an equation of the tangent to the parabola at \(P\) is \[py - x = ap^2\] (4)
(b) Write down the equation of the tangent at \(Q\). (1)
The tangent at \(P\) meets the tangent at \(Q\) at the point \(R\).
(c) Find, in terms of \(p\) and \(q\), the coordinates of \(R\), giving your answers in their simplest form. (4)
Given that \(R\) lies on the directrix of \(C\),
(d) find the value of \(pq\). (2)
| Scheme | Marks |
|---|---|
| \(y = 2a^{\frac{1}{2}}x^{\frac{1}{2}} \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = a^{\frac{1}{2}}x^{-\frac{1}{2}}\) \(y^2 = 4ax \Rightarrow 2y\dfrac{\mathrm{d}y}{\mathrm{d}x} = 4a\) or \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\mathrm{d}y}{\mathrm{d}t}.\dfrac{\mathrm{d}t}{\mathrm{d}x} = 2a.\dfrac{1}{2ap}\) \(x^{\frac{1}{2}} \to x^{-\frac{1}{2}}\) \(ky\dfrac{\mathrm{d}y}{\mathrm{d}x} = c\) \(\dfrac{\mathrm{d}y}{\mathrm{d}t} \times \dfrac{1}{\frac{\mathrm{d}x}{\mathrm{d}t}}\). Can be a function of \(p\) or \(t\). | M1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = a^{\frac{1}{2}}x^{-\frac{1}{2}}\) or \(2y\dfrac{\mathrm{d}y}{\mathrm{d}x} = 4a\) or \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2a.\dfrac{1}{2ap}\) Differentiation is accurate. | A1 |
| \(y - 2ap = \dfrac{1}{p}(x - ap^2)\) Applies \(y - 2ap = \text{their } m(x - ap^2)\) or \(y = (\text{their } m)x + c\) using \(x = ap^2\) and \(y = 2ap\) in an attempt to find \(c\). Their \(m\) must be a function of \(p\) from calculus. | M1 |
| \(py - x = ap^2\) * Correct completion to printed answer* | A1 cso |
| (4) |
| Scheme | Marks |
|---|---|
| \(qy - x = aq^2\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(qy - aq^2 = py - ap^2\) Attempt to obtain an equation in one variable \(x\) or \(y\) | M1 |
| \(y(q - p) = aq^2 - ap^2\) \(y = \dfrac{aq^2 - ap^2}{q - p}\) Attempt to isolate \(x\) or \(y\) | M1 |
| \(y = a(p + q)\) or \(ap + aq\) \(x = apq\) A1: Either one correct simplified coordinate A1: Both correct simplified coordinates | A1,A1 |
| \((R(apq,\ ap + aq))\) | |
| (4) |
| Scheme | Marks |
|---|---|
| \('apq' = -a\) Their \(x\) coordinate of \(R = -a\) | M1 |
| \(pq = -1\) Answer only: Scores 2/2 if \(x\) coordinate of \(R\) is \(apq\) otherwise 0/2. | A1 |
| (2) | |
| Total 11 |