Coordinate Geometry (Perpendicular bisector)

CIE, 2020, winter, paper 22, No 3

(a) Find the equation of the perpendicular bisector of the line joining the points $(12,1)$ and $(4,3)$, giving your answer in the form $y=m x+c$

(b) The perpendicular bisector cuts the axes at points $A$ and $B$. Find the length of $A B$.


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(a) $\begin{array}[t]{ll}& P=(12,1), Q=(4,3) \\&M=\text { midpoint }=\left(\dfrac{12+4}{2}, \dfrac{1+3}{2}\right)=(8,2) \\&\text { Gradient of } P Q=\dfrac{3-1}{4-12}=\dfrac{2}{-8}=-\dfrac{1}{4} \\&\text { Gradient of perpendicular line } =m=4 \\&\text{ Required line equation: } y-2=4(x-8) =4 x-32 \\&\therefore y =4 x-30\end{array}$

(b) When $x=0, y=-30$ and if $y=0, x=7.5.$ Thus $A(0,-30),B(7.5,0)$.

Length of $AB=\sqrt{(7.5-0)^2+(0-(-30))^2}=\dfrac{15\sqrt{17}}{2}$.
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