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  • Find the equation of each of the following parabolas (a) Directrix x = 0, focus at (6, 0) (b) Vertex at (0, 4), focus at (0, 2) (c) Focus at (–1, –2), directrix x – 2y + 3 = 0

Find the equation of each of the following parabolas

(a) Directrix x = 0, focus at (6, 0)

(b) Vertex at (0, 4), focus at (0, 2)

(c) Focus at (–1, –2), directrix x – 2y + 3 = 0

Answers (1)

We know that the distance of any point on the parabola from its focus and its directrix is same.

i) Given that directrix x=0 and focus (6,0)  

So, for any point P(x,y) on the parabola  

 Distance of P from directrix=Distance of P from focus x2=(x-6)2+y2

y2-12x+36=0 

 ii) Given that vertex=(0,4) and focus (0,2)

 Now distance between the vertex and directrix is same as the distance between the vertex and focus. 

 Distance of P from directrix=Distance of P from focus 

\left | y-6 \right |=\sqrt{\left ( x-0 \right )^{2}+\left ( y-2 \right )^{2}}

y2-12y+36=x2+y2-4y+4

x2=32-8y

iii) Given that focus is at (-1,-2)

and directrix x-2y+3=0

\sqrt{\left ( x+1 \right )^{2}+\left ( y+2 \right )^{2}}=\left | \frac{x-2y+3}{\sqrt{1+4}} \right |

x2+2x+1+y2+4y+4=1/5[x2+4y2+9+6x-4xy-12y]

4x2+4xy+y2+4x+32y+16=0

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