Given ns focus and also directrix of a parabola , how do we find the equation of the parabola?

If we consider only parabolas that open upwards or downwards, then the directrix will be a horizontal heat of the develop y = c .

Let ( a , b ) be the focus and lens y = c be the directrix. Lens ( x 0 , y 0 ) be any point on the parabola.

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Any point, ( x 0 , y 0 ) on the parabola satisfies the definition of parabola, so there are two distances to calculate:

Distance between the point on the parabola to the emphasis Distance between the point on the parabola to the directrix

To find the equation of the parabola, equate these two expressions and solve for y 0 .

Find the equation of the parabola in the example above.

Distance between the suggest ( x 0 , y 0 ) and ( a , b ) :

( x 0 − a ) 2 + ( y 0 − b ) 2

Distance between suggest ( x 0 , y 0 ) and the heat y = c :

|   y 0 − c   |

(Here, the distance between the point and horizontal line is difference of their y -coordinates.)

Equate the two expressions.

( x 0 − a ) 2 + ( y 0 − b ) 2 = |   y 0 − c   |

Square both sides.

( x 0 − a ) 2 + ( y 0 − b ) 2 = ( y 0 − c ) 2

Expand the expression in y 0 on both sides and simplify.

( x 0 − a ) 2 + b 2 − c 2 = 2 ( b − c ) y 0

This equation in ( x 0 , y 0 ) is true for all other values on the parabola and hence we can rewrite with ( x , y ) .

Therefore, the equation of the parabola with emphasis ( a , b ) and directrix y = c is

( x − a ) 2 + b 2 − c 2 = 2 ( b − c ) y




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Example:

If the focus of a parabola is ( 2 , 5 ) and the directrix ins y = 3 , find the equation of the parabola.

Lens ( x 0 , y 0 ) be any point on the parabola. Find the distance between ( x 0 , y 0 ) and the focus. Then find the distance in between ( x 0 , y 0 ) and directrix. Equate these two distance equations and the simplified equation in x 0 and also y 0 is equation of the parabola.

The distance between ( x 0 , y 0 ) and ( 2 , 5 ) is ( x 0 − 2 ) 2 + ( y 0 − 5 ) 2

The distance between ( x 0 , y 0 ) and the directrix, y = 3 is

|   y 0 − 3   | .

Equate the two distance expressions and square on both sides.

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( x 0 − 2 ) 2 + ( y 0 − 5 ) 2 = |   y 0 − 3   |

( x 0 − 2 ) 2 + ( y 0 − 5 ) 2 = ( y 0 − 3 ) 2

Simplify and bring all terms to one side:

x 0 2 − 4 x 0 − 4 y 0 + 20 = 0

Write the equation with y 0 on one side:

y 0 = x 0 2 4 − x 0 + 5

This equation in ( x 0 , y 0 ) is true for all other values on the parabola and hence we can rewrite via ( x , y ) .

So, the equation of the parabola with focus ( 2 , 5 ) and directrix ins y = 3 is