Question

Find the equation of the parabola with focus (2, 3) and directrix y = -1. A) (y - 1)2 = 8(x - 2) B) (y - 1)2 = 4(x - 2) C) (x - 2)2 = 8(y - 1) D) (x - 2)2 = 4(y - 1)

Answer 1

Answer:

The correct option is option (C).

The required equation of the parabola is

$(x-2)^2=8(y-1)$

Step-by-step explanation:

Formula:

• The distance of a point (x₁,y₁) from a line ax+by+c=0 is

$\frac{ax_1+by_1+c}{\sqrt{a^2+b^2}}$.

• The distance between two points (x₁,y₁) and (x₂,y₂) is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Given that, the focus of the parabola is (2,3) and directrix y= - 1.

We know that, a parabola is the locus of points that equidistant from the directrix and the focus.

Let any point on the parabola be  P(x,y)  .

The distance of P from the directrix is

$=\frac{y+1}{\sqrt {1^2}}$            

=y+1

The distance of the point P from the focus (2,3) is

$\sqrt{(x-2)^2+(y-3)^2}$

According to the problem,

$\sqrt{(x-2)^2+(y-3)^2}=y+1$

$\Rightarrow (x-2)^2+(y-3)^2=(y+1)^2$  

$\Rightarrow (x-2)^2+y^2-6y+9=y^2+2y+1$

$\Rightarrow (x-2)^2=y^2+2y+1-y^2+6y-9$

$\Rightarrow (x-2)^2=8y-8$

$\Rightarrow (x-2)^2=8(y-1)$

The required equation of the parabola is

$(x-2)^2=8(y-1)$