If the equation of the parabola, whose vertex is at (5, 4) and the directrix is 3x + y - 29 = 0, is x² + ay² + bxy + cx + dy + k = 0, then a + b + c + d + k is equal to :

Solution

Maths sir · 4 steps

Get the focus first, then distance to focus equals distance to directrix gives the whole equation.

1. Find the focus. The axis passes through the vertex (5, 4) along the normal direction (3, 1) of the directrix. The foot of the perpendicular from (5, 4) to 3x + y - 29 = 0 is (8, 5). The vertex is the midpoint of this foot and the focus, so the focus is S = (2, 3).
2. Focus-directrix property. Any point (x, y) on the parabola is as far from S as from the directrix. Square both distances and set them equal.

   (x-2)² + (y-3)² = (3x + y - 29)²/10

3. Expand. Multiply by 10: 10x² + 10y² - 40x - 60y + 130 = 9x² + y² + 6xy - 174x - 58y + 841. Bringing everything to one side gives x² + 9y² - 6xy + 134x - 2y - 711 = 0. The coefficient of x² is already 1, so this matches the given form.
4. So. a = 9, b = -6, c = 134, d = -2, k = -711. Then a + b + c + d + k = 9 - 6 + 134 - 2 - 711 = -576, option (d).

The idea to keep. Given the vertex and the directrix, get the focus by doubling the vertex-to-directrix step. Then distance to focus equals distance to directrix gives the whole equation.

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