Question

The curves x²− 4y² + c = 0 and y² = 4x will intersect orthogonally for

• A. c ∈ (0, 16)
• B. c ∈ (-3, 4)
• C. c ∈ (3, 4)
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Curves will intersect if x² - 16x + c = 0 has real roots.

Thus c ≤ 64. For x² − 4y² + c = 0, $\frac { d y } { d x } = \frac { x } { 4 y }$

For y² = 4x, $\frac { d y } { d x } = \frac { 2 } { y }$ . If curves intersect orthogonally then $\frac { x } { 4 y } \cdot \frac { 2 } { y }$ = −1

⇒ x = −2y² = −8x

⇒ x = 0 ⇒ y = 0. But if y = 0, slope of both curves are undefined.