If $\lambda x^2-10 x y+12 y^2+5 x-16 y-3=0$ represents a pair of straight lines, the value of $\lambda$ is:
a) -1
b) -3
c) -3
d) -7
Answer (1)
Substitute the coefficients of the given equation into the determinant condition for a pair of straight lines.
Expand the determinant: λ ( 12 ( − 3 ) − ( − 8 ) ( − 8 )) − ( − 5 ) (( − 5 ) ( − 3 ) − ( − 8 ) ( 2 5 )) + 2 5 (( − 5 ) ( − 8 ) − 12 ( 2 5 )) = 0 .
Simplify the equation: − 100 λ + 175 + 25 = 0 .
Solve for λ : λ = 2 .
Explanation
Problem Analysis We are given the equation λ x 2 − 10 x y + 12 y 2 + 5 x − 16 y − 3 = 0 , which represents a pair of straight lines. Our goal is to find the value of λ .
Identifying Coefficients The general equation of the second degree is given by a x 2 + 2 h x y + b y 2 + 2 gx + 2 f y + c = 0 . Comparing this with the given equation, we have: a = λ , 2 h = − 10 ⟹ h = − 5 , b = 12 , 2 g = 5 ⟹ g = 2 5 , 2 f = − 16 ⟹ f = − 8 , and c = − 3 .
Condition for Straight Lines For the equation to represent a pair of straight lines, the determinant of the matrix formed by the coefficients must be zero. That is, a h g h b f g f c = 0
Substituting Values Substituting the values, we get: λ − 5 2 5 − 5 12 − 8 2 5 − 8 − 3 = 0
Expanding the Determinant Expanding the determinant, we have: λ ( 12 ( − 3 ) − ( − 8 ) ( − 8 )) − ( − 5 ) (( − 5 ) ( − 3 ) − ( − 8 ) ( 2 5 )) + 2 5 (( − 5 ) ( − 8 ) − 12 ( 2 5 )) = 0 λ ( − 36 − 64 ) + 5 ( 15 + 20 ) + 2 5 ( 40 − 30 ) = 0 λ ( − 100 ) + 5 ( 35 ) + 2 5 ( 10 ) = 0 − 100 λ + 175 + 25 = 0 − 100 λ + 200 = 0
Solving for Lambda Solving for λ , we get: − 100 λ = − 200 λ = − 100 − 200 λ = 2
Final Answer Therefore, the value of λ is 2.
Examples
Understanding when a second-degree equation represents straight lines is useful in various fields, such as computer graphics and engineering. For example, in computer graphics, identifying lines helps in rendering images and creating geometric shapes. In engineering, it can be used to analyze structural stability and design layouts where linear elements are crucial. Knowing that λ = 2 ensures that the given equation models a pair of straight lines, which can then be used for further analysis or design purposes.
