The number of values of such that the straight line touches the curve is
- A.0
- B.2
- C.1
- D.
Show answer and solution
Correct option: B
Solution
For the line y = 4x + c to touch the ellipse x²/4 + y²= 1, substitute y = 4x + c into the ellipse equation and require the discriminant to be zero.
Substituting: x²/4 + (4x+c)² = 1 → x²/4 + 16x² + 8cx + c² = 1 → (65/4)x² + 8cx + (c²-1) = 0.
Setting discriminant = 0: (8c)² - 4·(65/4)·(c²-1) = 0 → 64c² - 65(c²-1) = 0 → -c² + 65 = 0 → c² = 65, giving exactly c = ±√65.
These are 2 distinct values of c.
Substituting: x²/4 + (4x+c)² = 1 → x²/4 + 16x² + 8cx + c² = 1 → (65/4)x² + 8cx + (c²-1) = 0.
Setting discriminant = 0: (8c)² - 4·(65/4)·(c²-1) = 0 → 64c² - 65(c²-1) = 0 → -c² + 65 = 0 → c² = 65, giving exactly c = ±√65.
These are 2 distinct values of c.
[1998]