In coordinate geometry, understanding how to find the gradient and equation of a line perpendicular to a given line is vital. This section focuses on the mathematical techniques for determining these with precision, tailored for CIE IGCSE students.

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Understanding Gradients

The gradient of a line indicates its steepness. For a line $y = mx + c$, $m$represents the gradient. Two lines are perpendicular if the product of their gradients equals $-1$.

Gradient of Perpendicular Lines

The gradient of a line perpendicular to another is the negative reciprocal of the original line's gradient. If a line has a gradient $m$, then the perpendicular line's gradient is $-\dfrac{1}{m}$.

Calculating the Gradient of a Perpendicular Line

Example 1: Gradient Calculation

Given $2y = 3x + 1$, find the gradient of a line perpendicular to it.

• Original line in slope form: $y = \frac{3}{2}x + \frac{1}{2}$
• Original gradient $(m)$ : $\frac{3}{2}$
• Perpendicular gradient $(m_{\text{perpendicular}})$: $-\frac{1}{\frac{3}{2}} = -\frac{2}{3}$

Finding the Equation of a Line Perpendicular to a Given Line

Given a point and the perpendicular gradient, use the formula $y - y_1 = m(x - x_1)$ to find the equation.

Example 2: Equation of a Perpendicular Line

Find the equation of a line perpendicular to $y = -2x + 5$ passing through $(3, 4)$.

• Perpendicular gradient: $\frac{1}{2}$
• Equation using point $(3, 4)$: $y - 4 = \frac{1}{2}(x - 3)$
• Simplified equation: $y = \frac{1}{2}x + 2.5$

Graphical Representation:

Practice Questions

Question 1:

Given the line equation $4x - y = 8$, calculate the gradient of a line perpendicular to it and find its equation if it passes through the point $(-1, -2)$.

Solution:

• Given line's gradient: $4$
• Perpendicular gradient: $-\frac{1}{4} = -0.25$
• Equation: $y - (-2) = -0.25(x - (-1))$
• Simplified equation: $y = -0.25x - 2.25$

Graphical Representation:

Question 2:

The line $y = \frac{4}{5}x - 3$ has a perpendicular line passing through the point $(2, 3)$. Determine the equation of this perpendicular line.

Solution:

• Given line's gradient: $\frac{4}{5}$
• Perpendicular gradient: $-\frac{5}{4} = -1.25$
• Equation: $y - 3 = -1.25(x - 2)$
• Simplified equation: $y = -1.25x + 5.5$

Graphical Representation: