To find the equation of line PQ, which is perpendicular to the line AB and passes through the point [tex]\( P(-2, 4) \)[/tex], follow these steps:
1. Find the slope of line AB:
The equation of line AB is given as [tex]\( 7x - 4y + 15 = 0 \)[/tex].
Rearrange it to the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
7x - 4y + 15 = 0 \Rightarrow -4y = -7x - 15 \Rightarrow y = \frac{7}{4}x + \frac{15}{4}
\][/tex]
So, the slope [tex]\( m \)[/tex] of line AB is [tex]\( \frac{7}{4} \)[/tex].
2. Determine the slope of line PQ:
Since line PQ is perpendicular to line AB, its slope will be the negative reciprocal of the slope of AB.
[tex]\[
m_{PQ} = -\frac{4}{7}
\][/tex]
3. Use the point-slope form of the equation of a line:
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( (x_1, y_1) \)[/tex] is a point on the line.
Here, [tex]\( (x_1, y_1) = (-2, 4) \)[/tex] and [tex]\( m = -\frac{4}{7} \)[/tex].
4. Substitute the point and the slope into the point-slope form:
[tex]\[
y - 4 = -\frac{4}{7}(x + 2)
\][/tex]
5. Simplify the equation:
[tex]\[
y - 4 = -\frac{4}{7}x - \frac{8}{7}
\][/tex]
[tex]\[
y = -\frac{4}{7}x - \frac{8}{7} + 4
\][/tex]
[tex]\[
y = -\frac{4}{7}x - \frac{8}{7} + \frac{28}{7}
\][/tex]
[tex]\[
y = -\frac{4}{7}x + \frac{20}{7}
\][/tex]
6. Express the equation in simplified form:
The equation of line PQ is:
[tex]\[
y = -\frac{4}{7}x + \frac{20}{7}
\][/tex]
Thus, the equation of the line PQ that passes through point [tex]\( P(-2, 4) \)[/tex] and is perpendicular to the line AB is:
[tex]\[
y = -\frac{4}{7}x + \frac{20}{7}
\][/tex]
