FP1 2017 3 Edexcel

3

(a) We have:

\displaystyle P = \left(4 \times \frac 1 4, 4 \times 4\right) = (1, 16)

and:

\displaystyle Q = \left(4 \times 2, \frac 4 2\right) = \left(8, 2\right)

The gradient of PQ is:

\displaystyle \frac {2 - 16} {8 - 1} = \frac {-14} 7 = -2

So the gradient of l is \dfrac 1 2. Since it passes through the origin:

\displaystyle y = \frac 1 2 x

(b)

A Cartesian equation for H is given by:

\displaystyle xy = 4t \times \frac 4 t = 16

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We have:

\displaystyle \frac 1 2 x^2 = 16

So:

\displaystyle x^2 = 32

giving:

x = \pm \sqrt {32} = \pm 4 \sqrt 2

If x = 4 \sqrt 2, then \displaystyle y = \frac 16 {4 \sqrt 2} = \frac 4 {\sqrt 2} = 2 \sqrt 2

Similarly if x = -4 \sqrt 2 then y = -2 \sqrt 2.

So the points that l intersects H are (\pm 4 \sqrt 2, \pm 2 \sqrt 2).

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