Answer:
The equation of a line perpendicular to another line can be found using the following steps:
1. Find the slope of the line AB:
The slope (m) of a line passing through points A = (-19, -2) and B = (3, -7) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Plugging in the values, we get:
m = (-7 - (-2)) / (3 - (-19))
= (-7 + 2) / (3 + 19)
= -5 / 22
2. Find the negative reciprocal of the slope:
The negative reciprocal of the slope (-5/22) is obtained by flipping the fraction and changing the sign. So, the negative reciprocal is 22/5.
3. Use the slope-intercept form of the equation (y = mx + c):
We know that the line passes through point C, which divides the line AB in the ratio 2:5. Since point C is nearer to point A, we can assume that point C is 2/7 times the distance from point A to point B.
To find the coordinates of point C, we can use the following formula:
x-coordinate of C = (2 * x-coordinate of B + 5 * x-coordinate of A) / (2 + 5)
y-coordinate of C = (2 * y-coordinate of B + 5 * y-coordinate of A) / (2 + 5)
Plugging in the values, we get:
x-coordinate of C = (2 * 3 + 5 * (-19)) / (2 + 5)
= (6 - 95) / 7
= -89/7
y-coordinate of C = (2 * (-7) + 5 * (-2)) / (2 + 5)
= (-14 - 10) / 7
= -24/7
Now, we have the coordinates of point C, which are (-89/7, -24/7), and the negative reciprocal of the slope, which is 22/5.
Substituting these values into the slope-intercept form of the equation (y = mx + c), we get:
y = (22/5) * x + c
4. Find the value of c:
To find the value of c, we can substitute the coordinates of point C into the equation and solve for c. Using the coordinates (-89/7, -24/7), we get:
-24/7 = (22/5) * (-89/7) + c
Solving this equation, we find:
c = -24/7 - (22/5) * (-89/7)
= -24/7 + 22 * 89/35
= -24/7 + 1958/35
= -240/70 + 1958/35
= -240/70 + 1114/70
= 874/70
= 437/35
Therefore, the equation of the line perpendicular to line AB and passing through point C is:
y = (22/5) * x + 437/35