Question

If for some p, q, r ∈ R, not all have same sign, one of the roots of the equation (p² + q²)x² – 2q(p + r)x + q² + r² = 0 is also a root of the equation x² + 2x – 8 = 0, then (q² + r²)/p² is equal to _______ .

Answer 1

Correct Answer: 272

Step 1: Understand the Given Equation

We are given that one of the roots of the equation (p² + q²)x² - 2pqx + r² = 0 is also a root of the equation x² + 2x - 8 = 0. We need to find the ratio (q² + r²) / p².

Step 2: Solve the Second Equation

Let's start by solving the quadratic equation x² + 2x - 8 = 0 to find its roots. Using the quadratic formula:

The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

For the equation x² + 2x - 8 = 0, we have:

• a = 1, b = 2, c = -8

Substitute these values into the formula:

x = (-2 ± √(2² - 4(1)(-8))) / 2(1)

x = (-2 ± √(4 + 32)) / 2

x = (-2 ± √36) / 2

x = (-2 ± 6) / 2

Thus, the roots are:

• x₁ = 2, x₂ = -4

Step 3: Using the Roots to Set Up the First Equation

We know that one of the roots of (p² + q²)x² - 2pqx + r² = 0 is either x₁ = 2 or x₂ = -4. Let's substitute these roots into the first equation.

For the root x₁ = 2, substitute into the equation:

(p² + q²)(2)² - 2pq(2) + r² = 0

(p² + q²)(4) - 4pq + r² = 0

4(p² + q²) - 4pq + r² = 0

Step 4: Solve for the Desired Ratio

We have now the equation 4(p² + q²) - 4pq + r² = 0. From here, we can proceed with solving for the ratio of (q² + r²) / p².

We are given that the equation x² + 2x - 8 = 0 shares roots with the first equation. Therefore, solving for the ratio involves simplifying the equation step by step.

After manipulating the equation, we find:

The ratio (q² + r²) / p² = √[ (2 + 3 sinθ) ]

Conclusion

The ratio (q² + r²) / p² is equal to √[ (2 + 3 sinθ) ], where θ is the angle provided in the context of the problem.

Answer: √[ (2 + 3 sinθ) ]