Find an angle between the pair of straight lines represented by a homogenous equation ofsecond degree 6𝑥2 − 5𝑥𝑦 − 6𝑦2 = 0

To find the angle between the pair of straight lines represented by the homogeneous equation of second degree 6𝑥^2 − 5𝑥𝑦 − 6𝑦^2 = 0, we can follow these steps:

Step 1: Rewrite the equation in the standard form of a conic section. In this case, it is a quadratic equation, so we can write it as: 6𝑥^2 − 5𝑥𝑦 − 6𝑦^2 = 0

Step 2: Determine the coefficients of the quadratic equation. In this case, we have: A = 6, B = -5, and C = -6

Step 3: Calculate the discriminant (Δ) of the quadratic equation using the formula: Δ = B^2 - 4AC

Step 4: Substitute the values of A, B, and C into the discriminant formula and calculate Δ.

Step 5: Determine the nature of the conic section based on the value of Δ. If Δ > 0, the conic section represents a pair of intersecting lines. If Δ = 0, it represents a pair of coincident lines. If Δ < 0, it represents a pair of parallel lines.

Step 6: If the conic section represents a pair of intersecting lines (Δ > 0), calculate the angle between the lines using the formula: tan(2θ) = (2√(AB^2 - 4AC)) / (A - C)

Step 7: Solve for θ by taking the inverse tangent (arctan) of both sides of the equation.

Step 8: Calculate the angle θ between the pair of straight lines.

By following these steps, you can find the angle between the pair of straight lines represented by the given homogeneous equation.