Question

A line AB passes through the point P(3,-2) with gradient -1/2, determine the equation of the line CD through P perpendicular to AB.

Answer 1

Answer on Question #83036 – Math – Analytic Geometry

Question

line AB passes through the point $P(3, -2)$ with gradient $-1/2$, determine the equation of the line CD through P perpendicular to AB.

Solution

AB: $y = k_1x + b_1$

CD: $y = k_2x + b_2$

CD: $y - y_0 = k_2(x - x_0)$

It is given that CD is perpendicular to AB $\Rightarrow k_2 = -\frac{1}{k_1}$, where $k_1$ is the gradient of AB and $k_2$ is the gradient of CD;

$$k_1 = -\frac{1}{2} \Rightarrow k_2 = -\frac{1}{-\frac{1}{2}} = 2$$

Besides, CD goes through the point $P(3, -2)$. Thus, $k_2 = 2$, $x_0 = 3$, $y_0 = -2$. Finally,

CD: $y - (-2) = 2(x - 3)$

$$\begin{array}{l} y + 2 = 2x - 6 \\ y = 2x - 6 - 2 \\ y = 2x - 8 \\ \end{array}$$

Answer: $y = 2x - 8$.

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