Question

If the line x – 1 = 0 is the directrix of the parabola

y² – kx + 8 = 0, k ¹ 0 and the parabola intersects the circle

x² + y² = 4 in two real distinct points, then the value of k is-

• A. –4
• B. –8
• C. 4
• D. None of these

Answer 1

Answer: (B)

–8

Answer 2

The equation of the parabola can be written as

y² = k(x – 8/k) which is of the form Y² = 4AX

where Y = y, X = x – 8/k and A = k/4

Equation of the directrix is X = –A Ž x – 8/k = –k/4

Which represents the given line x – 1 = 0 if $\frac{8}{k} - \frac{k}{4} = 1$

Ž k² + 4k – 32 = 0 Ž k = –8 or 4

For k = 4, the parabola is y² = 4(x – 2) whose vertex is (2, 0) and touches the circle

x² + y² = 4 at the vertex. Therefore k ¹ 4 .

For k = –8, the parabola is y² = –8(x + 1) which intersects the circle x² + y² = 4 at two real distinct points as the vertex (–1, 0) of the parabola lies inside the circle.