### Locus

Question Sample Titled 'Locus'

Consider two fixed points ${A}{\left({0},-{6}\right)}$ and ${B}{\left({0},{8}\right)}$ on a rectangular plane.

 (a) A moving point ${P}$ is always equidistant from the two fixed points. Denote the locus of ${P}$ by ${L}$. Find the equation of ${L}$. (2 marks) (b) Consider another moving point ${Q}$ such that ${Q}{A}$ is always perpendicular to ${Q}{B}$ except at ${A}$ or ${B}$. ${A}$ and ${B}$ are two possible positions of ${Q}$ . (4 marks) (i) Find the equation of the locus of ${Q}$. (ii) Describe the locus of ${Q}$.

 (a) Let ${\left({x},{y}\right)}$ be the coordinates of ${P}$. ${P}{A}$ $={P}{B}$ $\sqrt{{{\left({x}-{0}\right)}^{{2}}+{\left({y}-{\left(-{6}\right)}\right)}^{{2}}}}$ $=\sqrt{{{\left({x}-{0}\right)}^{{2}}+{\left({y}-{\left({8}\right)}\right)}^{{2}}}}$ 1M ${y}$ $={1}$ $\therefore$The equation of ${L}$ is ${y}={1}$ . 1A  (bi) Let the coordinates of ${Q}$ be ${\left({x},{y}\right)}$ . ${(}$Slope of ${Q}{A}{)}{(}$Slope of ${Q}{B}{)}$ $=-{1}$ 1M ${\left(\dfrac{{{y}-{\left(-{6}\right)}}}{{{x}-{0}}}\right)}{\left(\dfrac{{{y}-{\left({8}\right)}}}{{{x}-{0}}}\right)}$ $=-{1}$ 1M ${x}^{{2}}+{y}^{{2}}-{2}{y}-{48}$ $={0}$ $\therefore$The equation of the locus of ${Q}$ is ${x}^{{2}}+{y}^{{2}}-{2}{y}-{48}={0}$ 1A (bii) The locus of ${Q}$ is a circle with centre ${\left({0},{1}\right)}$ and radius ${7}$ units. 1A

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