Answer:
The distance from the given point to the given line is 5√17.
Explanation:
Recall that the shortest distance from a point to a line is on another line, which in turn is perpendicular to the given line. Here the given line is y = 4x + 27, and the slope is 4; any line perpendicular to this line has a slope which is the negative reciprocal of 4, which is -1/4.
Find the equation of the line with slope -1/4 passing through the given point (12, -10):
Use the slope-intercept method: y = mx + b becomes -10 = (-1/4)(12) + b. Find the y-intercept, b: -10 = -3 + b. then b = -7.
The perpendicular line is y = (-1/4)x - 7.
Now we want to find the distance between this given point (12, -10) to the intersection of the perpendicular line y = (-1/4)x - 7 with the given line y = 4x + 27. In other words we must solve the system of linear equations:
y = (-1/4)x - 7
y = 4x + 27
and to do this we equate these two equations, eliminating y and enabling us to find the x-coordinate of the point of intersection of the two lines.
y = 4x + 27 = y = (-1/4)x - 7
This becomes (4 1/4)x = -34, or
17x -34
----- = -------
4 1
Cross multiplying, we get 17x = - 136, or x = -8.
Since y = 4x + 27, if x = -8 we get y = 4(-8) + 27, or -32 + 27, or -5.
The point of intersection of the two lines is then (-8, -5).
All we need to do now is to find the distance between (-8, -5) and the given point (12, -10). The change in x as we go from (-8, -5) to (12, -10) is 20 and the change in y is -5. The distance we want is found using the Pythagorean Theorem: (20)^2 + (-5)^2 = d^2, where d represents that distance;
we get d^2 = 400 + 25, or d^2 = 425, or d = +√425, or
d = +√(25)(17), or d +5√17.
The distance from the given point to the given line is 5√17.