Question

Let y = 4x² and $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{16} = 1$ intersect iff

• A. |a| ≤$\frac{1}{\sqrt{2}}$
• B. a < - $\frac{1}{\sqrt{2}}$
• C. a> - $\frac{1}{\sqrt{2}}$
• D. None of these

Answer 1

Answer: (A)

|a| ≤$\frac{1}{\sqrt{2}}$

Answer 2

y = 4x² and $\frac{1}{4}$y = x²

using $\frac{1}{4a^{2}}y - \frac{y^{2}}{16} = 1$

⇒ 4y – a²y² = 16a²

⇒ a²y² – 4y + 16a² = 0

⇒ D ≥ 0 for intersection of two curves

⇒ 16 – 4a² (16a²) ≥ 0

⇒ 1 – 4a⁴ ≥ 0

⇒ $\left( \sqrt{2}a^{2} \right)^{2} \leq 1$

⇒ $|\sqrt{2}a| \leq 1$

⇒ $- \frac{1}{\sqrt{2}} \leq a \leq \frac{1}{\sqrt{2}}$