### Straight lines, locus and circle

Question Sample Titled 'Straight lines, locus and circle'

The ${y}$-intercept of two parallel lines ${L}$ and ${l}$ are $-{1}$ and $-{3}$ respectively and the ${x}$-intercept of ${L}$ is ${3}$ . ${S}$ is a moving point in the rectangular coordinate plane such that the perpendicular distance from ${S}$ to ${L}$ is equal to the perpendicular distance from ${S}$ to ${l}$ . Denote the locus of ${S}$ by $\Gamma$ .

 (a) (i) Describe the geometric relationship between $\Gamma$ and ${L}$ . (5 marks) (ii) Find the equation of $\Gamma$ . (b) The equation of circle ${C}$ is ${\left({x}-{6}\right)}^{{2}}+{y}^{{2}}={4}$ . Denote the centre of ${C}$ by ${G}$ . (4 marks) (i) Does $\Gamma$ pass through ${G}$ ? Explain your answer. (ii) If ${L}$ cuts ${C}$ at ${A}$ and ${B}$ while $\Gamma$ cuts ${C}$ at ${H}$ and ${K}$ , find the ratio of the area of $\triangle{A}{G}{H}$ to the area of $\triangle{B}{G}{K}$ .

 (ai) $\Gamma$ is parallel to ${L}$ . 1A (aii) Note that the ${y}$-intercept of $\Gamma$ is $-{2}$ . 1A The slope of ${L}$ $=\dfrac{{-{1}-{0}}}{{{0}-{3}}}$ 1M  $=\dfrac{{1}}{{3}}$ 1A The equation of $\Gamma$ is ${y}+{2}$ $=\dfrac{{1}}{{3}}{\left({x}-{0}\right)}$ ${x}-{3}{y}-{6}$ $={0}$ 1A  (bi) Note that the coordinates of ${G}$ are ${\left({6},{0}\right)}$ . 1A Since ${\left({6}\right)}-{3}{\left({0}\right)}-{6}$ $={0}$ , $\Gamma$ passes through ${G}$ . 1A  (bii) Note that both ${G}{H}$ and ${G}{K}$ are radii of the circle. 1M Also note that both the heights of $\triangle{A}{G}{H}$ and $\triangle{B}{G}{K}$ are the distance between ${L}$ and $\Gamma$ . ∴  The area of $\triangle{A}{G}{H}$ is equal to the area of $\triangle{B}{G}{K}$ . Thus, the required ratio is ${1}:{1}$ . 1A

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