Question

In what ratio is the line segment joining A(8, 9) and B(-7, 4) divided by

(a) the point (2, 7) the x-axis the y-axis.

Answer 1

(a) 2:3 internally, (b) 9:4 externally, (c) 8:7 internally

Answer 2

The problem requires finding the ratio in which a line segment joining two points A(8, 9) and B(-7, 4) is divided by:

(a) a given point (2, 7) (b) the x-axis (c) the y-axis

We will use the section formula. If a point P(x, y) divides the line segment joining A(x1, y1) and B(x2, y2) in the ratio k:1, then the coordinates of P are given by:

$x = \frac{k x_2 + 1 x_1}{k+1}$ $y = \frac{k y_2 + 1 y_1}{k+1}$

If k is positive, the division is internal. If k is negative, the division is external, and the ratio is |k|:1.

Given points: A(x1, y1) = (8, 9) and B(x2, y2) = (-7, 4).

Part (a): Division by the point (2, 7)

Let the point P(2, 7) divide the line segment AB in the ratio k:1. Using the x-coordinate section formula:

$2 = \frac{k(-7) + 1(8)}{k+1}$ $2(k+1) = -7k + 8$ $2k + 2 = -7k + 8$ $9k = 6$ $k = \frac{6}{9} = \frac{2}{3}$

Since k is positive, the division is internal. The ratio is k:1 = $\frac{2}{3}:1$, which simplifies to 2:3. (We can verify with the y-coordinate: $7 = \frac{\frac{2}{3}(4) + 1(9)}{\frac{2}{3}+1} = \frac{\frac{8}{3} + 9}{\frac{5}{3}} = \frac{\frac{8+27}{3}}{\frac{5}{3}} = \frac{35/3}{5/3} = \frac{35}{5} = 7$. This confirms the ratio.)

Part (b): Division by the x-axis

A point on the x-axis has its y-coordinate equal to 0. Let the point of division be P(x, 0). Let this point divide the line segment AB in the ratio k:1. Using the y-coordinate section formula:

$0 = \frac{k(4) + 1(9)}{k+1}$ $0 = 4k + 9$ $4k = -9$ $k = -\frac{9}{4}$

Since k is negative, the division is external. The ratio is $|k|:1 = \frac{9}{4}:1$, which simplifies to 9:4 externally.

Part (c): Division by the y-axis

A point on the y-axis has its x-coordinate equal to 0. Let the point of division be P(0, y). Let this point divide the line segment AB in the ratio k:1. Using the x-coordinate section formula:

$0 = \frac{k(-7) + 1(8)}{k+1}$ $0 = -7k + 8$ $7k = 8$ $k = \frac{8}{7}$

Since k is positive, the division is internal. The ratio is k:1 = $\frac{8}{7}:1$, which simplifies to 8:7.