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The points P and Q have coordinates (-1, 6) and (9, 0) respectively.
The line l is perpendicular to PQ and passes through the mid-point of PQ.
Find an equation for l, giving your answer in the form ax + by + c =0, where a, b and c are integers.

Respuesta :

MrRoyal

Answer:

[tex]3y - 5x +11=0[/tex]

Step-by-step explanation:

Given

[tex]P(x_1,y_1) = (-1,6)[/tex]

[tex]Q(x_2,y_2) = (9,0)[/tex]

Required

The equation of l

First, calculate the slope (m) of PQ

[tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]

[tex]m = \frac{0-6}{9--1}[/tex]

[tex]m = \frac{-6}{10}[/tex]

[tex]m = \frac{-3}{5}[/tex]

Since l is perpendicular to PQ, the slope of l is:

[tex]m_2 = -\frac{1}{m}[/tex]

[tex]m_2= -\frac{1}{-3/5}[/tex]

[tex]m_2 = \frac{5}{3}[/tex]

Next, calculate the midpoint of PQ

[tex]M = \frac{1}{2}(x_1 + x_2,y_1+y_2)[/tex]

[tex]M = \frac{1}{2}(-1+9,6+0)[/tex]

[tex]M = \frac{1}{2}(8,6)[/tex]

[tex]M = (4,3)[/tex]

The equation of l is:

[tex]y = m(x -x_1) + y_1[/tex]

[tex]y = \frac{5}{3}(x -4) +3[/tex]

Multiply through by 3

[tex]3y = 5(x -4) +9[/tex]

Open bracket

[tex]3y = 5x -20 +9[/tex]

[tex]3y = 5x -11[/tex]

Rewrite as:

[tex]3y - 5x +11=0[/tex]

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