If [tex]$\lambda x^2-10 x y+12 y^2+5 x-16 y-3=0$[/tex] represents a pair of straight lines, the value of [tex]$\lambda$[/tex] is:
a) -1
b) -3
c) -3
d) -7
Answer (1)
The problem requires finding the value of λ for which the given equation represents a pair of straight lines.
Compare the given equation with the general second-degree equation to identify the coefficients.
Apply the condition that the determinant of the matrix formed by the coefficients must be zero.
Solve the resulting equation for λ , which gives 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 given 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 of a , h , b , g , f , and c , 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
The concept of representing a pair of straight lines with a second-degree equation is used in various fields like computer graphics and structural engineering. For instance, in computer graphics, detecting intersections of lines is a fundamental operation, and understanding the conditions under which a second-degree equation represents straight lines helps in developing efficient algorithms for collision detection. In structural engineering, analyzing stress distributions often involves identifying lines of maximum stress, which can be modeled using similar mathematical techniques. These applications highlight the practical relevance of understanding the properties of second-degree equations and their geometric interpretations.
