Answer:
[tex]24x+7y+20=0[/tex]
[tex]\overline{PQ}=4[/tex]
Step-by-step explanation:
The line PQ is perpendicular to the line 7x - 24y + 10 = 0 and passes through point P(-2, 4).
To find the equation of line PQ, we first need to find its slope.
Since the slopes of perpendicular lines are negative reciprocals of each other, and the slope of line 7x - 24y + 10 = 0 is 7/24, the slope of line PQ is:
[tex]m=-\dfrac{24}{7}[/tex]
Now, substitute the found slope m = -24/7 and the point (-2, 4) into the point-slope formula to write the equation of line PQ:
[tex]\begin{aligned}y-y_1&=m(x-x_1)\\\\y-4&=-\dfrac{24}{7}(x-(-2))\\\\y-4&=-\dfrac{24}{7}(x+2)\\\\7(y-4)&=-24(x+2)\\\\7y-28&=-24x-48\\\\24x+7y-28+48&=0\\\\24x+7y+20&=0\end{aligned}[/tex]
Therefore, the equation of line PQ is:
[tex]24x+7y+20 = 0[/tex]
Rearrange to isolate y:
[tex]y=-\dfrac{24}{7}x-\dfrac{20}{7}[/tex]
Point Q is the point of intersection of line PQ and line 7x - 24y + 10 = 0.
To determine the length of PQ, we need to find the coordinates of point Q. To find the x-coordinate of point Q, substitute the equation for line PQ into the equation for line 7x - 24y + 10 = 0:
[tex]\begin{aligned}7x-24\left(-\dfrac{24}{7}x-\dfrac{20}{7}\right)+10&=0\\\\7x+\dfrac{576}{7}x+\dfrac{480}{7}\right)+10&=0\\\\\dfrac{625}{7}x+\dfrac{550}{7}\right)&=0\\\\625x+550&=0\\\\625x&=-550\\\\x&=-\dfrac{550}{625}\\\\x&=-\dfrac{22}{25}\end{aligned}[/tex]
Now, substitute the x-coordinate into the equation of line PQ to find the y-coordinate of point Q:
[tex]\begin{aligned}y&=-\dfrac{24}{7}\left(-\dfrac{22}{25}\right)-\dfrac{20}{7}\\\\y&=\dfrac{528}{175}-\dfrac{20}{7}\\\\y&=\dfrac{528}{175}-\dfrac{500}{175}\\\\y&=\dfrac{28}{175}\\\\y&=\dfrac{4}{25}\end{aligned}[/tex]
Therefore, the coordinates of point Q are:
[tex]Q\left(-\dfrac{22}{25},\dfrac{4}{25}\right)[/tex]
Finally, to determine the length of PQ, we can use the distance formula:
[tex]\boxed{\begin{array}{l}\underline{\sf Distance \;Formula}\\\\d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\\textsf{where:}\\ \phantom{ww}\bullet\;\;d\;\textsf{is the distance between two points.} \\\phantom{ww}\bullet\;\;\textsf{$(x_1,y_1)$ and $(x_2,y_2)$ are the two points.}\end{array}}[/tex]
Substitute the coordinates of points P and Q into the distance formula:
[tex]\begin{aligned}\overline{PQ}&=\sqrt{\left(x_Q-x_P\right)^2+\left(y_Q-y_P\right)^2}\\\\\overline{PQ}&=\sqrt{\left(-\dfrac{22}{25}-(-2)\right)^2+\left(\dfrac{4}{25}-4\right)^2}\\\\\overline{PQ}&=\sqrt{\left(\dfrac{28}{25}\right)^2+\left(-\dfrac{96}{25}\right)^2}\\\\\overline{PQ}&=\sqrt{\dfrac{784}{625}+\dfrac{9216}{625}}\\\\\overline{PQ}&=\sqrt{\dfrac{10000}{625}}\\\\\overline{PQ}&=\dfrac{100}{25}\\\\\overline{PQ}&=4\end{aligned}[/tex]
Therefore, the length of PQ is 4 units.
