Question

Obtain eqñ of parabola with focus(3,2)
Directrix 3x-4y+9=0

Answer 1

Answer on Question #80501 – Math – Analytic Geometry

Question

Obtain eqñ of parabola with focus (3,2)

Directrix 3x-4y+9=0

Solution

The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus.

The distance from point $(x,y)$ to the point $(3,2)$ is $\sqrt{(x - 3)^2 + (y - 2)^2}$

The distance from point $(x,y)$ to the line $3x - 4y + 9 = 0$ is $\frac{|3x - 4y + 9|}{\sqrt{3^2 + 4^2}}$

These distances must be equal. The equation will be

$$\sqrt{(x - 3)^2 + (y - 2)^2} = \frac{|3x - 4y + 9|}{\sqrt{3^2 + 4^2}}$$

After squaring both sides

$$(x - 3)^2 + (y - 2)^2 = \frac{(3x - 4y + 9)^2}{25}$$ $$25(x^2 - 6x + 9 + y^2 - 4y + 4) = 9x^2 + 16y^2 - 24xy + 54x - 72y + 81$$

from which

$$16x^2 + 9y^2 + 24xy - 204x - 28y + 244 = 0$$

Answer: $16x^2 + 9y^2 + 24xy - 204x - 28y + 244 = 0$

Answer

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