Question

Find the value of lambda. ​

Answer 1

EXPLANATION.

⇒ λx² - 10xy + 12y² + 5x - 16y - 3 = 0.

Represent a pair of straight lines.

As we know that,

General equation of straight line of second degree.

⇒ ax² + 2hxy + by² + 2gx + 2fy + c = 0.

Compare the equation with general equation, we get.

⇒ λ = a, h = -5, b = 12, g = 5/2, f = -8, c = -3

As we know that,

⇒ abc + 2fgh - af² - bg² - ch² = 0.

Put the value in the equation, we get.

⇒ (λ)(12)(-3) + 2(-8)(5/2)(-5) - (λ)(-8)² - (12)(5/2)² - (-3)(-5)² = 0.

⇒ -36λ + 200 - 64λ² - 75 + 75 = 0.

⇒ -36λ + 200 - 64λ = 0.

⇒ -100λ + 200 = 0.

⇒ 100λ = 200.

⇒ λ = 2.

                                                                                                                       

MORE INFORMATION.

Homogeneous equation.

(1) = If y = m₁x and y = m₂x be the two equation represent by ax² + 2hxy + by² = 0 then,

m₁ + m₂ = -2h/b, m₁m₂ = a/b.

(2) = If θ is the acute angle between the pair of straight lines then,

tanθ = |2√h² - ab/a + b|.

(3) = The equation of the straight lines bisecting the angle between the straight lines,

ax² + 2hxy + by² = 0 is x² - y²/a - b = xy/h.