Porisma

Worked solution

Question 5

The graph of (x−p)2+(y−p)2=p2(x-p)^2+(y-p)^2=p^2, p>0p>0, underwent exactly two simple transformations, involving either reflection, stretching or translation only, to become the ellipse that passes through (0,p)(0,p), (2p,p)(2p,p), (p,3p)(p,3p) and (p,−p)(p,-p).

(i) Write down the equation of the ellipse. marks

(ii) Describe clearly (in words) a sequence of the two transformations. mark

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Target ellipse centred at (p, p) for p > 0, through (0, p), (2p, p), (p, 3p), (p, −p). The horizontal semi-axis is p and vertical semi-axis 2p.
Target ellipse centred at (p, p) for p > 0, through (0, p), (2p, p), (p, 3p), (p, −p). The horizontal semi-axis is p and vertical semi-axis 2p.
Target ellipse centred at (p, p) for p > 0, through (0, p), (2p, p), (p, 3p), (p, −p). The horizontal semi-axis is p and vertical semi-axis 2p.

2010/RI/Promo/5(b)

Answer

(i) (x−p)2p2+(y−p)2(2p)2=1\dfrac{(x-p)^2}{p^2}+\dfrac{(y-p)^2}{(2p)^2}=1.

(ii) Scale the graph parallel to the yy-axis by a factor of 2, then translate it by pp units in the negative yy-direction. Equivalently, translate it by p2\dfrac p2 units in the negative yy-direction first, then scale parallel to the yy-axis by a factor of 2.

Compare your reasoning

The ellipse has centre (p,p)(p,p), half-width pp and half-height 2p2p.

Scaling the circle by a factor of 2 parallel to the yy-axis moves its centre to (p,2p)(p,2p) and gives the half-height 2p2p. Translating down by pp returns the centre to (p,p)(p,p). Reversing the order, the translation must be p2\dfrac p2 because the scaling then doubles it.

How did this question go?

Compare your sketch

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Target ellipse centred at (p, p) for p > 0, through (0, p), (2p, p), (p, 3p), (p, −p). The horizontal semi-axis is p and vertical semi-axis 2p.
Target ellipse centred at (p, p) for p > 0, through (0, p), (2p, p), (p, 3p), (p, −p). The horizontal semi-axis is p and vertical semi-axis 2p.
Target ellipse centred at (p, p) for p > 0, through (0, p), (2p, p), (p, 3p), (p, −p). The horizontal semi-axis is p and vertical semi-axis 2p.