Question

1. a) point p (1, 0) partitions \\(\overline{mn}\\) in the ratio of 5:4. the coordinates of m are (-4, 5). find the coordinates of n.

Explanation:

Step1: Recall the section formula

The section formula for a point \( P(x, y) \) that divides the line segment joining \( M(x_1, y_1) \) and \( N(x_2, y_2) \) in the ratio \( m:n \) is given by:

$$ x=\frac{mx_2 + nx_1}{m + n}, \quad y=\frac{my_2 + ny_1}{m + n} $$

Here, \( P(1, 0) \), \( M(-4, 5) \), ratio \( m:n = 5:4 \), and we need to find \( N(x_2, y_2) \).

Step2: Solve for \( x_2 \)

Substitute the known values into the \( x \)-coordinate formula:

$$ 1=\frac{5x_2 + 4(-4)}{5 + 4} $$

Simplify the denominator: \( 5 + 4 = 9 \)

$$ 1=\frac{5x_2 - 16}{9} $$

Multiply both sides by 9:

$$ 9 = 5x_2 - 16 $$

Add 16 to both sides:

$$ 5x_2 = 9 + 16 = 25 $$

Divide by 5:

$$ x_2=\frac{25}{5}=5 $$

Step3: Solve for \( y_2 \)

Substitute the known values into the \( y \)-coordinate formula:

$$ 0=\frac{5y_2 + 4(5)}{5 + 4} $$

Simplify the denominator: \( 5 + 4 = 9 \)

$$ 0=\frac{5y_2 + 20}{9} $$

Multiply both sides by 9:

$$ 0 = 5y_2 + 20 $$

Subtract 20 from both sides:

$$ 5y_2=-20 $$

Divide by 5:

$$ y_2=\frac{-20}{5}=-4 $$

Answer:

The coordinates of \( N \) are \( (5, -4) \)