| Date | May 2007 | Marks available | 4 | Reference code | 07M.1.hl.TZ0.1 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Show that | Question number | 1 | Adapted from | N/A |
Question
The point \({\rm{P}}(x,y)\) moves in such a way that its distance from the point (\(1\) , \(0\)) is three times its distance from the point (\( -1\) , \(0\)) .
Find the equation of the locus of P.
Show that this equation represents a circle and state its radius and the coordinates of its centre.
Markscheme
We are given that
\(\sqrt {{{(x - 1)}^2} + {y^2}} = 3\sqrt {{{(x + 1)}^2} + {y^2}} \) M1A1
\({x^2} - 2x + 1 + {y^2} = 9({x^2} + 2x + 1 + {y^2})\) A1
\(8{x^2} + 8{y^2} + 20x + 8 = 0\) A1
[4 marks]
Rewrite the equation in the form
\({\left( {x + \frac{5}{4}} \right)^2} + {y^2} = - 1 + \frac{{25}}{{16}} = \frac{9}{{16}}\) M1A1
This represents a circle with radius \( = \frac{3}{4}\) ; centre \(\left( { - \frac{5}{4},0} \right)\) A1A1
Note: Allow FT from the line above.
[4 marks]
