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

Given:
A(–1, 7), B(4, –3)
P divides AB in the ratio 3 : 2.

Step 1: Find coordinates of point P
Formula for internal division:
\[ P(x, y) = \left( \frac{m_2 x_1 + m_1 x_2}{m_1+m_2},\ \frac{m_2 y_1 + m_1 y_2}{m_1+m_2} \right) \]
Here, \(m_1 = 3\), \(m_2 = 2\).

\[ x = \frac{2(-1) + 3(4)}{5} = \frac{-2 + 12}{5} = \frac{10}{5} = 2 \] \[ y = \frac{2(7) + 3(-3)}{5} = \frac{14 - 9}{5} = \frac{5}{5} = 1 \]
So, \[ P(2,\ 1) \]

Step 2: Verify that P lies on the line \( x - 3y = -1 \)
Substitute x = 2, y = 1:
\[ 2 - 3(1) = 2 - 3 = -1 \]
The equation is satisfied.

Therefore, \[ \boxed{P(2,1)\ \text{lies on the line}\ x - 3y = -1} \]

Step 3: Find lengths PA and PB
Distance formula:
\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]

Length PA:
A(–1, 7), P(2, 1)
\[ PA = \sqrt{(2 + 1)^2 + (1 - 7)^2} \] \[ = \sqrt{3^2 + (-6)^2} \] \[ = \sqrt{9 + 36} \] \[ = \sqrt{45} = 3\sqrt{5} \]
Length PB:
P(2, 1), B(4, –3)
\[ PB = \sqrt{(4 - 2)^2 + (-3 - 1)^2} \] \[ = \sqrt{2^2 + (-4)^2} \] \[ = \sqrt{4 + 16} \] \[ = \sqrt{20} = 2\sqrt{5} \]

Final Answers:
• Point of division: \( P(2, 1) \)
• P lies on the line \( x - 3y = -1 \)
• \( PA = 3\sqrt{5} \) units
• \( PB = 2\sqrt{5} \) units