Q) 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.

(Q 26 B – 30/5/2 – CBSE 2026 Question Paper)

Ans:

(i) Verifying if P lies on line segment AB:
[Approach: To solve this, we first need to find the coordinates of point P. Since division ratio is given, we can calculate these coordinates using the section formula. Next, we will verify if it satisfies the given line equation, and finally calculate the distances PA and PB.]

Step 1; By section formula for a point dividing the segment AB in the ratio m : n is:

P(x, y) =

Now, we are given points, A (-1, 7) and B (4, -3), ratio given, m : n = 3 : 2

Step 2: Coordinates of point P:

x:

= = 2

y:

= = 1

Hence, coordinates of point P are (2, 1).

Step 3: If P lies on the line x – 3 y = – 1, then it will satisfy the line equation.

Let’s put the value of x and y in the equation’s LHS and check if it meets RHS or not.

∴ LHS: (2) – 3 (1)

= 2 – 3 = – 1 = RHS

∵ LHS = RHS, Therefore, the point P (2, 1) lies on the line.

(ii) Calculation of lengths of PA and PB:

Step 4: We know that according to the distance formula, distance between 2 points (x₁,y₁) and (x₂,y₂)

D =

Length of PA between A (-1, 7) and P (2, 1):

PA =

= √(9 + 36) = √45 = 3√5 units

Length of PB between P(2, 1) and B(4, -3):

PB =

= √(4 + 16) = √20 = 2√5 units

Therefore the length of PA and PB are 3√5 units and 3√5 units

Check: If PA = 3√5, PB = 2√5
∴ PA : PB = 3√5 : 2√5 = 3:2
Since it meets the given condition, our solution is correct.

Please press “Heart” if you liked the solution.