Question

It is given that a straight line has the equation (y = 2x + 6), and a point (P(12, 10)). Find the coordinates of (Q) such that it is on the straight line (y = 2x + 6), and (PQ) is the shortest distance.

a) (Q(9, 18))
b) (Q(7, 14))
c) (Q(10, 16))
d) (Q(15, 26))

Answer 1

To find point Q on the line y = 2x + 6 closest to P(12, 10), we determine the perpendicular slope to the line, set up an equation for the line perpendicular to y = 2x + 6 through P, and find their intersection.

Step-by-step explanation:

The problem asks us to find the point Q that lies on the line y = 2x + 6 and is closest to point P(12, 10). The shortest distance from a point to a line is along the perpendicular to the line that passes through the point in question.

First, we need the slope of the given line, which is the coefficient of x in the equation y = 2x + 6. Therefore, the slope is 2. The perpendicular slope is the negative reciprocal, so we use -1/2. Now we write the equation of the line that is perpendicular to y = 2x + 6 and passes through P(12, 10).

The equation of the perpendicular line is y - 10 = -1/2 (x - 12). Solving this simultaneously with the original line equation will give us point Q. Substituting x from one equation to the other, we can solve for y and subsequently find x, thus getting the coordinates of Q, which will be one of the given options and the shortest distance PQ will be the perpendicular distance.