188
Gurveer and Arushi built a robot that can paint a path as it moves on a graph paper. Some co-ordinate of points are marked on it. It starts from $(0, 0)$, moves to the points listed in order (in straight lines) and ends at $(0, 0)$.
Arushi entered the points $P(8, 6), Q(12, 2)$ and $S(-6, 6)$ in order. The path drawn by robot is shown in the figure.
Based on the above, answer the following questions :
(i) Determine the distance OP.
(ii) QS is represented by equation $2x + 9y = 42$. Find the co-ordinates of the point where it intersects $y$-axis.
(iii) (a) Point $R(4.8, y)$ divides the line segment OP in a certain ratio, find the ratio. Hence, find the value of $y$.
OR
(iii) (b) Using distance formula, show that $\frac{PQ}{OS} = \frac{2}{3}$.
Arushi entered the points $P(8, 6), Q(12, 2)$ and $S(-6, 6)$ in order. The path drawn by robot is shown in the figure.
Based on the above, answer the following questions :
(i) Determine the distance OP.
(ii) QS is represented by equation $2x + 9y = 42$. Find the co-ordinates of the point where it intersects $y$-axis.
(iii) (a) Point $R(4.8, y)$ divides the line segment OP in a certain ratio, find the ratio. Hence, find the value of $y$.
OR
(iii) (b) Using distance formula, show that $\frac{PQ}{OS} = \frac{2}{3}$.
Show SolutionHide Solution↓
(i) The distance $OP = \sqrt{64 + 36} = 10$ (1 mark)
(ii) $2x + 9y = 42$ intersects $y$-axis at $(0, \frac{14}{3})$ (1 mark)
(iii) (a) Let $OR : RP = k : 1$, therefore $4.8 = \frac{8k}{k + 1} \Rightarrow k = \frac{3}{2} \Rightarrow OR : RP = 3 : 2$ ($1\frac{1}{2}$ marks)
$y = \frac{18}{5}$ ($\frac{1}{2}$ mark)
OR
(iii) (b) $PQ = \sqrt{4^2 + (-4)^2} = \sqrt{32} \text{ or } 4\sqrt{2}$ ($\frac{1}{2}$ mark)
$OS = \sqrt{(-6)^2 + 6^2} = \sqrt{72} \text{ or } 6\sqrt{2}$ ($\frac{1}{2}$ mark)
$\therefore \frac{PQ}{OS} = \sqrt{\frac{32}{72}} = \sqrt{\frac{4}{9}} = \frac{2}{3}$ (1 mark)
(ii) $2x + 9y = 42$ intersects $y$-axis at $(0, \frac{14}{3})$ (1 mark)
(iii) (a) Let $OR : RP = k : 1$, therefore $4.8 = \frac{8k}{k + 1} \Rightarrow k = \frac{3}{2} \Rightarrow OR : RP = 3 : 2$ ($1\frac{1}{2}$ marks)
$y = \frac{18}{5}$ ($\frac{1}{2}$ mark)
OR
(iii) (b) $PQ = \sqrt{4^2 + (-4)^2} = \sqrt{32} \text{ or } 4\sqrt{2}$ ($\frac{1}{2}$ mark)
$OS = \sqrt{(-6)^2 + 6^2} = \sqrt{72} \text{ or } 6\sqrt{2}$ ($\frac{1}{2}$ mark)
$\therefore \frac{PQ}{OS} = \sqrt{\frac{32}{72}} = \sqrt{\frac{4}{9}} = \frac{2}{3}$ (1 mark)