### angle in circle, equation of straight line

Question Sample Titled 'angle in circle, equation of straight line'

In the figure, ${O}$ , ${C}$ , ${D}$ and ${M}$ are four points on a circle. ${B}$ and ${A}$ are two points on the ${x}$-axis such that ${D}{B}$ and ${C}{A}$ are perpendicular to the ${x}$-axis. It is given that $\angle{C}{D}{M}$ is a right angle and $\angle{O}{C}{D}={45}^{\circ}$ .

 (a) Find $\angle{M}{O}{B}$ . (2 marks) (b) Find the equation of ${O}{M}$ . (3 marks) (c) If the coordinates of ${D}$ are ${\left({18},{s}\right)}$ , where ${s}$ is a constant, find the coordinates of ${M}$ . (1 mark)

${x}$${y}$${C}$${D}$${M}$${O}$${B}$${A}$${45}^{\circ}$

 (a) $\angle{O}{M}{B}$ $=\angle{O}{C}{D}={45}^{\circ}$ ext. ∠, cyclic quad. 1A In $\triangle{M}{O}{B}$ , $\angle{M}{O}{B}$ $={180}^{\circ}-{45}^{\circ}-{90}^{\circ}$ ∠ sum of △  $={45}^{\circ}$ 1A  (b) Slope of ${O}{M}$ $={\tan}\angle{M}{O}{B}$  $={{\tan{{45}}}^{\circ}}$ 1M  $={1}$  Equation of ${O}{M}$ is: $\dfrac{{{y}-{0}}}{{{x}-{0}}}$ $={1}$ 1M ${y}$ $={x}$ 1A  (c) ∵  ${D}{M}{B}\bot{x}$-axis. ∴   The ${x}$-coordinate of ${M}=$The ${x}$-coordinate of ${D}$  $={18}$  Put ${x}={18}$ into the equation of ${O}{M}$, we have ${y}={18}$ . Thus, the coordinates of ${M}$ are ${\left({18},{18}\right)}$ . 1A

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