Question

If roots of the equation 3x² + 5x + 1 = 0 are

(sec θ₁ – tan θ₁) and (cosec θ₂ – cot θ₂), then the equation whose roots are (sec θ₂ + tan θ₂) and (cosec θ₂ + cot θ₂) will be –

• A. 3x² + 5x + 1 = 0
• B. x² + 5x + 3 = 0
• C. 3x² – 9x + 7 = 0
• D. 7x² – 9x + 2 = 0

Answer 1

Answer: (B)

x² + 5x + 3 = 0

Answer 2

Let secθ₁ – tanθ₁ = α, then secθ₁ + tanθ₁ = $\frac{1}{\alpha}$ and cosecθ₂ – cotθ₂ = β, then cosec θ₂ + tanθ₂ = $\frac{1}{\beta}$ given

α + β = – $\frac{5}{3}$,

αβ = $\frac{1}{3}$

Required equation will be x² – $\left( \frac{1}{\alpha} + \frac{1}{\beta} \right)$ x + $\frac{1}{\alpha\beta}$ = 0

⇒ x² – $\left( \frac{\alpha + \beta}{\alpha\beta} \right)$ x + $\frac{1}{\alpha\beta}$ = 0

⇒ x² + 5x + 3 = 0