QQuestion
A and B are two points on the x-axis and y-axis respectively, as shown in the figure.
(a) Write down the coordinates of A and B.
(b) P is a point on AB such that AP : PB = 3 : 1. Using the section formula, find the coordinates of P.
(c) Find the equation of a line passing through P and perpendicular to AB.
✓Answer
(a) From the figure:
A = (4, 0) and B = (0, 4).
(b) P divides AB internally in the ratio AP : PB = 3 : 1.
Using the section formula:
P=\left(\frac{m_1x_2+m_2x_1}{m_1+m_2},\frac{m_1y_2+m_2y_1}{m_1+m_2}\right) P=\left(\frac{3(0)+1(4)}{3+1},\frac{3(4)+1(0)}{3+1}\right) P=\left(\frac{4}{4},\frac{12}{4}\right)Therefore, P = (1, 3).
(c) Slope of AB:
m_{AB}=\frac{4-0}{0-4} m_{AB}=\frac{4}{-4}=-1The product of the slopes of perpendicular lines is −1.
(−1) × m = −1
Therefore, m = 1.
The required line passes through P(1, 3).
Using y − y1 = m(x − x1):
y − 3 = 1(x − 1)
y − 3 = x − 1
Therefore, y = x + 2.
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