Question

If P( 9a - 2, - b ) divides the line segment joining A( 3a + 1, - 3 ) and B( 8a, 5 ) in the ratio 3 : 1. Find the value of a and b. Also find the value of a^2 + b^2.

Answer 1

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SOLUTION :-
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To find the values of a and b we need to use the sectional formula, which is m₁x₂ + m₂x₁ / m₁+m₂ , m₁y₂ + m₂y₁ / m₁+m₂

P(x) = m₁x₂ + m₂x₁ / m₁+m₂
⇒ 9a-2 = 3(8a) + 1(3a+1)
⇒ 4(9a-2) = 24a +3a+1
⇒ 36a-8 = 27a +1
⇒ 9a = 9
⇒ a=1
∴ The value of a is 1

P(y) = m₁y₂ + m₂y₁ / m₁+m₂
⇒ -b = 3(5) + 1(-3) / 3+1
⇒ -b = 15 -3 /4
⇒ -b = 12/4
⇒ -b = 3
⇒ b = -3
∴The value of b = -3

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Answer 2

Answer:

a = 1, b = -3, a² + b² = 10

Step-by-step explanation:

Given P(9a-2, -b) divides A(3a + 1,-3) and B(8a,5) in ratio 3:1.

Here, m = 3, n = 1

Section formula:

⇒ (mx₂ + nx₁/m + n, m₂y₁ + m₁y₂/m + n) = (9a-2,b)

⇒ (3 * 8a + 3a + 1/3 + 1,  3 * 5 - 3/3 + 1) = (9a - 2,b)

⇒ (27a + 1/4, 3) = (9a - 2,b)

(i)

⇒ 27a + 1/4 = 9a - 2

⇒ 27a + 1 = 4(9a - 2)

⇒ 27a + 1 = 36a -8

⇒ a = 1

(ii)

-b = 3

⇒ b = -3.

Now,

⇒ a² + b²

⇒ 1 + 9

⇒ 10.

Therefore, a = 1,b = -3, a² + b² = 10.

Hope it helps!