Given points P(−1,−2) and Q(4,2) in the cartesian plane, point R(1,n) is taken so that the sum of the distances PR+RQ is the smallest it can be. What is the value of n?
- n=−0.6,
- n=−0.4,
- n=−0.2,
- n=0.2,
- n=±0.2.
Bonus: draw a clear diagram illustrating your solution.
Solution
If PR+RQ is at its smallest, then R must be some point on the line PQ, so let’s start by finding the equation y=mx+b of that line. First the slope:
m=△x△y=4−(−1)2−(−2)=54.
So far the equation is y=0.8x+b. To find b, we plug in a point. We’ll use Q.
y=0.8x+b→2=0.8(4)+b→2=3.2+b→b=−1.2.
Therefore, the line PQ has the equation y=0.8x−1.2.
To find R, we simply plug in 1 to the equation, and our y-value will be n:
y=0.8x−1.2→n=0.8(1)−1.2→n=0.8−1.2→n=−0.4.
Point R is therefore (1,−0.4) and the correct answer is b.