Question

Prove that the point P dividing the line segment joining the points A(-1, 7) and B(4, -3) in the ratio 3 : 2, lies on the line \(x - 3y = -1\). Also find length of PA and PB.

Hint:

If you find the ratio of lengths \(PA/PB\), it must equal the given section ratio (3:2). In this case, \(3\sqrt{5} / 2\sqrt{5} = 3/2\). This is a great way to verify your answer!

Answer 1

Step 1: Understanding the Concept:
We use the section formula to find the coordinates of point \(P\). Then, we substitute these coordinates into the given line equation to verify if it satisfies the equation. Finally, we use the distance formula for \(PA\) and \(PB\).
Step 2: Key Formula or Approach:
Section Formula: \(P(x, y) = \left(\frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n}\right)\)
Distance Formula: \(d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}\)
Step 3: Detailed Explanation:
Find coordinates of \(P\) with \(m=3, n=2, A(-1, 7), B(4, -3)\):
\[ x = \frac{3(4) + 2(-1)}{3 + 2} = \frac{12 - 2}{5} = \frac{10}{5} = 2 \]
\[ y = \frac{3(-3) + 2(7)}{3 + 2} = \frac{-9 + 14}{5} = \frac{5}{5} = 1 \]
So, \(P\) is \((2, 1)\).
Verification: Substitute \(P(2, 1)\) into \(x - 3y = -1\):
L.H.S. \(= 2 - 3(1) = 2 - 3 = -1\).
Since L.H.S. \(=\) R.H.S., point \(P\) lies on the line.
Calculating lengths:
\[ PA = \sqrt{(2 - (-1))^{2} + (1 - 7)^{2}} = \sqrt{3^{2} + (-6)^{2}} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} \text{ units} \]
\[ PB = \sqrt{(4 - 2)^{2} + (-3 - 1)^{2}} = \sqrt{2^{2} + (-4)^{2}} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \text{ units} \]
Step 4: Final Answer:
Point \(P(2, 1)\) satisfies the equation \(x - 3y = -1\), proving it lies on the line.
The lengths are \(PA = 3\sqrt{5}\) units and \(PB = 2\sqrt{5}\) units.