A straight path from \(P(-4,\,7)\) to \(Q(6,\,-3)\) crosses the y-axis at the point \(R\). What is \(PR:RQ\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2 : 3
Every point on the y-axis has x-coordinate 0. So the section formula for the x-coordinate, with the unknown ratio \(k:1\), is set equal to 0.
Let R divide PQ in the ratio \(k:1\), with \(P(-4,\,7)\) and \(Q(6,\,-3)\)
x-coordinate of R \(\displaystyle =\frac{k(6)+1(-4)}{k+1}=\frac{6k-4}{k+1}\)
On the y-axis: \(6k-4=0\), so \(\displaystyle k=\frac{4}{6}=\frac{2}{3}\)
Ratio \(\displaystyle =\frac{2}{3}:1=2:3\)
Check: y-coordinate \(\displaystyle =\frac{2(-3)+3(7)}{2+3}=\frac{15}{5}=3\), so \(R=(0,\,3)\)
