If the equation of the parabola with vertex and the directrix x + 2y = 0 is , then i equal to : [2025]

Question

If the equation of the parabola with vertex $V (\frac{3}{2}, 3)$ and the directrix x + 2y = 0 is $α x^{2} + β y^{2} - γ x y - 30 x - 60 y + 225 = 0$ , then $α + β + γ$ is equal to : [2025]

Smart Achievers · Accepted Answer

(1)

Given : Vertex of parabola $(\frac{3}{2}, 3)$ and directrix is x + 2y = 0

Since, axis is $⊥$ to directrix and passes through vertex, then equation of axis

$y - 3 = 2 (x - \frac{3}{2}) \Rightarrow y = 2 x - 3 + 3 \Rightarrow y = 2 x$

$∴$ Foot of directrix is intersecting point of

y = 2x & 2y + x = 0 i.e., (0, 0)

$∴$ Focus $\equiv$ (3, 6)

Using definition of parabola,

$P S^{2} = P M^{2}$

$\Rightarrow {(x - 3)}^{2} + {(y - 6)}^{2} = {(\frac{x + 2 y}{\sqrt{5}})}^{2}$

$\Rightarrow x^{2} + 9 - 6 x + y^{2} + 36 - 12 y = \frac{x^{2} + 4 y^{2} + 4 x y}{5}$

$\Rightarrow 4 x^{2} + y^{2} - 4 x y - 30 x - 60 y + 225 = 0$

On comparing we get $α = 4, β = 1 and γ = 4$

Hence, $α + β + γ$ = 4 + 1 + 4 = 9.

Solution

Q.
If the equation of the parabola with vertex $V (\frac{3}{2}, 3)$ and the directrix x + 2y = 0 is $α x^{2} + β y^{2} - γ x y - 30 x - 60 y + 225 = 0$ , then $α + β + γ$ is equal to : [2025]

1 9

2 6

3 8

4 7

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Solution

Q. If the equation of the parabola with vertex V(32,3) and the directrix x + 2y = 0 is αx2+βy2–γxy–30x–60y+225=0, then α+β+γ is equal to : [2025]

1 9

2 6

3 8

4 7

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Q.
If the equation of the parabola with vertex $V (\frac{3}{2}, 3)$ and the directrix x + 2y = 0 is $α x^{2} + β y^{2} - γ x y - 30 x - 60 y + 225 = 0$ , then $α + β + γ$ is equal to : [2025]