Question 15:
(a) Diagram 11 shows a line segment AC.
It is given that point B divides the line segment AC in the ratio of m : n.
$$ \text { Show that } y=\frac{n y_1+m y_2}{m+n} \text {. } $$
(b) $$ \text { It is given that } P(x, y) \text { moves such that its distance from } $$
$$ R\left(\frac{3}{2},-\frac{1}{2}\right) \text { is } 2.5 \text { times its distance from the y-axis } $$
$$ \text { Find the equation of locus of } P \text {. } $$
(c) Diagram 12 shows the points P, Q, R and S on a Cartesian plane.
It is given that W (x, y) moves such that its distance from point P and point R is always equal.
If the area of the quadrilateral PQRS is 60 unit2, find the y-intercept of the straight line PQ.
[3 marks]
Answer:
(a) $$ \begin{aligned} \frac{y_2-y}{y-y_1} & =\frac{n}{m} \\ m y_2-m y & =n y-n y_1 \\ m y_2+n y_1 & =n y+m y \\ y(n+m) & =m y_2+n y_1 \\ y & =\frac{n y_1+m y_2}{m+n} \end{aligned} $$
(b) $$ \begin{aligned} P R & =2.5 x \\ \sqrt{\left(x-\frac{3}{2}\right)^2+\left[y-\left(-\frac{1}{2}\right)\right]^2} & =2.5 x \\ \left(\frac{2 x-3}{2}\right)^2+\left(\frac{2 y+1}{2}\right)^2 & =\left(\frac{5 x}{2}\right)^2 \\ \frac{4 x^2-12 x+9}{4}+\frac{4 y^2+4 y+1}{4} & =\frac{25 x^2}{4} \\ -21 x^2-12 x+4 y^2+4 y+10 & =0 \end{aligned} $$
(c) $$ \begin{aligned} & \frac{1}{2}\left|\begin{array}{cccc} 2 & 8 & 0 & 2 \\ 12 & 9 & y & 12 \end{array}\right|=30 \\ & \mid 2(9)+8 y+0(12)-12(8)-9(0)-2 y \mid=60 \\ &|18+8 y-96-2 y|=60 \\ & 18+8 y-96-2 y=60 \\ & y=23(\text { Ignore }) \end{aligned} $$
$$ \begin{aligned} &\text { or }\\ &\begin{aligned} 18+8 y-96-2 y & =-60 \\ y & =3 \end{aligned} \end{aligned} $$
(a) Diagram 11 shows a line segment AC.
It is given that point B divides the line segment AC in the ratio of m : n.$$ \text { Show that } y=\frac{n y_1+m y_2}{m+n} \text {. } $$
(b) $$ \text { It is given that } P(x, y) \text { moves such that its distance from } $$
$$ R\left(\frac{3}{2},-\frac{1}{2}\right) \text { is } 2.5 \text { times its distance from the y-axis } $$
$$ \text { Find the equation of locus of } P \text {. } $$
(c) Diagram 12 shows the points P, Q, R and S on a Cartesian plane.
It is given that W (x, y) moves such that its distance from point P and point R is always equal.If the area of the quadrilateral PQRS is 60 unit2, find the y-intercept of the straight line PQ.
[3 marks]
Answer:
(a) $$ \begin{aligned} \frac{y_2-y}{y-y_1} & =\frac{n}{m} \\ m y_2-m y & =n y-n y_1 \\ m y_2+n y_1 & =n y+m y \\ y(n+m) & =m y_2+n y_1 \\ y & =\frac{n y_1+m y_2}{m+n} \end{aligned} $$
(b) $$ \begin{aligned} P R & =2.5 x \\ \sqrt{\left(x-\frac{3}{2}\right)^2+\left[y-\left(-\frac{1}{2}\right)\right]^2} & =2.5 x \\ \left(\frac{2 x-3}{2}\right)^2+\left(\frac{2 y+1}{2}\right)^2 & =\left(\frac{5 x}{2}\right)^2 \\ \frac{4 x^2-12 x+9}{4}+\frac{4 y^2+4 y+1}{4} & =\frac{25 x^2}{4} \\ -21 x^2-12 x+4 y^2+4 y+10 & =0 \end{aligned} $$
(c) $$ \begin{aligned} & \frac{1}{2}\left|\begin{array}{cccc} 2 & 8 & 0 & 2 \\ 12 & 9 & y & 12 \end{array}\right|=30 \\ & \mid 2(9)+8 y+0(12)-12(8)-9(0)-2 y \mid=60 \\ &|18+8 y-96-2 y|=60 \\ & 18+8 y-96-2 y=60 \\ & y=23(\text { Ignore }) \end{aligned} $$
$$ \begin{aligned} &\text { or }\\ &\begin{aligned} 18+8 y-96-2 y & =-60 \\ y & =3 \end{aligned} \end{aligned} $$