Question

Point P is the foot of perpendicular from $A( - 5,7)$ to the line $2x - 3y + 18 = 0.$ Determine
A) The equation of the line AP
B) The coordinate of P.

Answer 1

Hint: According to given in the question we have to determine (i) the equation of the line AP and (ii) the coordinate of P. So, first of all we have to rearrange the terms of the expression which is $2x - 3y + 18 = 0$ as given in the question so we have to take y in the left and side and the remaining integers and numbers to the right hand side of the expression to determine the value of slope which is (m).
Now, to determine the slope we have to compare the obtained expression with the general form of the expression which is mentioned below:
$ \Rightarrow y = mx + c..................(A)$
Where, m is the slope and c is the constant term or any integer.
Now, we have to find the equation of the slope with the help of the point $A( - 5,7)$ and the slope of the line and as mentioned in the question that Point P is the foot of perpendicular from $A( - 5,7)$ to the line $2x - 3y + 18 = 0.$ So, we have to find the slope for the perpendicular line which can be determined with the help of the formula as mentioned below:

Formula used:
$
   \Rightarrow M = - \dfrac{1}{m}..................(B) \\
   \Rightarrow (y - {y_1}) = m(x - {x_1})............(C) \\
 $
Where, m is the slope and $({x_1},{y_1})$ are the points.
Now, to determine the coordinate of P we have to solve the expression obtained with the expression as mentioned in the question.

Complete step-by-step answer:
Step 1: First of all we have to rearrange the terms of the expression which is $2x - 3y + 18 = 0$ as given in the question so we have to take y in the left and side and the remaining integers and numbers to the right hand side of the expression to determine the value of slope which is (m) as mentioned in the solution hint. Hence,
$ \Rightarrow y = \dfrac{2}{3}x + 6$……………….(1)
Step 2: Now, to obtain the value of m which is the slope can be determined by comparing the expression with the expression (A) as mentioned in the solution hint. Hence,
$ \Rightarrow m = \dfrac{2}{3}$
Step 3: Now, to obtain the expression we have to determine the slope for the perpendicular line which can be determined with the help of the (B) which is as mentioned in the solution hint. Hence,
$ \Rightarrow M = - \dfrac{3}{2}$
Step 4: Now, we have to determine the equation with the help of the point $A( - 5,7)$ and with the help of the slope as we have obtained in the solution step 3 with the help of the formula (C) as mentioned in the solution hint. Hence,
\[
   \Rightarrow (y - 7) = - \dfrac{3}{2}(x - ( - 5)) \\
   \Rightarrow (y - 7) = - \dfrac{3}{2}(x + 5) \\
 \]
Now, to solve the expression we have to apply the cross-multiplication in the expression as obtained just above,
\[
   \Rightarrow 2y - 14 = - 3x - 15 \\
   \Rightarrow 2y + 3x - 14 + 15 = 0 \\
   \Rightarrow 3x + 2y + 1 = 0 \\
 \]
Step 5: Now, we have to solve both of the expressions which are given in the question and the expressions we obtained in the solution hint. So, we have to multiply with 3 to the expression obtained which is \[3x + 2y + 1 = 0\] and then we have to add it with the expression \[2 \times (2x - 3y + 18) = 0.\]Hence,
$
   \Rightarrow 3(3x + 2y + 1) + 2(2x - 3y + 18) = 0 \\
   \Rightarrow 9x + 6y + 3 + 4x - 6y + 36 = 0 \\
   \Rightarrow 13x = - 39 \\
   \Rightarrow x = - \dfrac{{39}}{{13}} \\
   \Rightarrow x = - 3 \\
 $
Step 6: Now, to obtain the value of y we have to substitute the value of x as obtained in the solution step 5 in the expression \[3x + 2y + 1 = 0\]. Hence,
$
   \Rightarrow 3( - 3) + 2y + 1 = 0 \\
   \Rightarrow - 9 + 2y + 1 = 0 \\
   \Rightarrow 2y = 8 \\
   \Rightarrow y = \dfrac{8}{2} \\
   \Rightarrow y = 4 \\
 $

Hence, with the help of the formulas (B) and (C) we have determine (i) the equation of the line AP which is \[3x + 2y + 1 = 0\] and (ii) the coordinate of point P which is $P( - 3,4)$.

Note:
To determine the expression of line AP it is necessary that we have to determine the slope of the perpendicular line and as we know that the line is perpendicular hence, to slope can be determine with the help of the formula $M = - \dfrac{1}{m}.$
To determine the coordinates it is necessary that we have to determine the equation of the line AP and then we have to solve this with the equation which is already mentioned in the question.