Question

A movable parabola touches the x and the y-axes at (1, 0) and (0, 1). Then the locus of the focus of the parabola is-

• A. 2x² – 2x + 2y² – 2y + 1 = 0
• B. x² – 2x + 2y² – 2y + 1 = 0
• C. 2x² – 2x + 2y² + 2y + 2 = 0
• D. 2x² + 2x – 2y² – 2y – 2 = 0

Answer 1

Answer: (A)

2x² – 2x + 2y² – 2y + 1 = 0

Answer 2

Since the x-axis and the y-axis are two perpendicular tangents to the parabola and both meet at the origin, the directix passes through the origin.

Let y = mx be the direction and (h, k) be the focus.

FA = AM

Ž $\sqrt{(h - 1)^{2} + k^{2}}$ = $\left| \frac{m}{\sqrt{1 + m^{2}}} \right|$ … (1)

and FB = BN

$\sqrt{h^{2} + (k - 1)^{2}}$= $\left| \frac{1}{\sqrt{1 + m^{2}}} \right|$ … (2)

From equation (1) and (2), we get

(h – 1)² + h² + k² + (k – 1)² = 1

Ž 2x² – 2x + 2y² – 2y + 1 = 0 is the required locus.

Hence (1) is correct answer.