Let P₁ be a parabola with vertex (3, 2) and focus (4, 4) and P₂ be its mirror image with respect to the line x + 2y = 6. Then the directrix of P₂ is x + 2y = ____________.

Solution

Maths sir · 4 steps

Find the directrix of P₁ first, then reflect that line across x + 2y = 6.

1. Axis direction. The axis runs from the vertex (3, 2) to the focus (4, 4), so its direction is (1, 2). The directrix is perpendicular to the axis, so it has the form x + 2y = c.
2. Directrix of P1. The vertex is the midpoint of the focus and the foot of the directrix on the axis. That foot is (3, 2) - (1, 2) = (2, 0). Putting (2, 0) into x + 2y gives c = 2, so the directrix of P₁ is x + 2y = 2.
3. Reflect the line. The line x + 2y = 2 is parallel to the mirror x + 2y = 6. Its image is another parallel line, at the same distance on the other side of the mirror. So the constant moves from 2 to 6 + (6 - 2).

   c' = 2 × 6 - 2 = 10

4. So. The directrix of P₂ is x + 2y = 10, so the blank is 10.

The common slip: 2. Stopping at the directrix of P₁ gives 2, not 10. The question asks for the directrix after reflecting it in x + 2y = 6.

The idea to keep. A mirror image of a parabola has the mirror image of its directrix. If that directrix is parallel to the mirror line ax + by = k, the image of ax + by = c is ax + by = 2k - c.

Answer from the NTA final answer key. Solution by PrepWiser.