If the equation of the parabola with vertex V(3/2, 3) and the directrix x+2 y=0 is α x²+β y²-γ x y-30 x-60 y+225=0, then α+β+γ is equal to :

Solution

Maths sir · 5 steps

Drop a perpendicular from the vertex to the directrix, then reflect it to get the focus.

1. Foot on directrix. The axis passes through V(3/2, 3) perpendicular to x + 2y = 0, so it runs along (1, 2). The foot of the perpendicular is Z = V - t(1,2) with t = (3/2 + 6)/(1² + 2²) = 3/2, which gives Z = (0, 0).
2. Find the focus. The vertex is the midpoint of Z and the focus S. So S = 2V - Z = (3, 6).
3. Focus-directrix rule. Every point on the parabola is as far from S as from the directrix. Squaring both distances gives the equation below.

   (x-3)² + (y-6)² = (x+2y)²/5

4. Expand. Multiply by 5: 5x² + 5y² - 30x - 60y + 225 = x² + 4xy + 4y². Rearranging gives 4x² + y² - 4xy - 30x - 60y + 225 = 0. The -30x, -60y and 225 match the given form, which confirms the focus.
5. So. α = 4, β = 1, γ = 4, so α + β + γ = 9, option (d).

The idea to keep. For a tilted parabola, find the focus first: the vertex is the midpoint of the focus and the foot of the perpendicular on the directrix. Then use PS² = (distance to directrix)².

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