Math, asked by ganapo7223, 11 months ago

If the two lines x + (a - 1)y = 1 and 2x + a²y = 1 (a ∈ R - {0,1}) are perpendicular, then the distance of their point of intersection from the origin is:
(A) 2/5
(B) √2/5
(C) 2/√5
(D) √(2/5)

Answers

Answered by abhi178
2

it is given that the two lines x + (a - 1)y = 1 and 2x + a²y = 1 (a ∈ R - {0,1}) are perpendicular.

we have to find distance of their point of intersection from origin.

slope of first line, m₁ = -1/(a - 1)

slope of 2nd line , m₂ = -2/a²

m₁ × m₂ = -1

⇒-1/(a - 1) × -2/a² = -1

⇒2/a²(a - 1) = -1

⇒2 = -a³ + a²

⇒a³ - a² + 2 = 0

⇒a³ + a² - 2a² + 2 = 0

⇒a²(a + 1) -2(a - 1)(a + 1) = 0

⇒(a + 1)(a² - 2a + 2) = 0

a = -1 , a² - 2a + 2 ≠ 0

so first line is x + (-1 - 1)y = 1 ⇒x - 2y = 1 ....(1)

2nd line is 2x + (-1)²y = 1 ⇒2x + y = 1......(2)

from equations (1) and (2) we get,

4x + x = 3 ⇒x = 3/5

so, y = 1 - 2x = 1 - 6/5 = -1/5

so, distance from origin of (3/5, -1/5) is √{(3/5-0)²+(-1/5-0)²} = √{9/25 + 1/25} = √{10/25} =√(2/5)

therefore, the distance of their point of intersection from the origin is √(2/5).

option (D) is correct choice.

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