Question

Vertex and focus of a parabola are (-1,1) and (2,3) respectively . Find the equation of the directrix .
(Please solve it by using foot of the perpendicular)

Answer 1

Answer:

3x + 2y + 14 = 0

Step-by-step explanation:

Let the line through the focus F and the vertex V (this is the axis of the parabola) meet the directrix in the point P (i.e. P is the foot of the perpendicular from V to the directrix).

As V is the midpoint of FP:

P - V = V - F

=>  P = 2V - F  =  (  2×(-1) - 2 ,  2×1 - 3 )  =  ( -4, -1 )

The slope of the axis FV is

( 3 - 1 ) / ( 2 - -1 ) = 2 / 3.

The directrix is perpendicular to the axis, so its slope is -3 / 2 and its equation is

3x + 2y + c = 0

for some c.

Since the point P = ( -4, -1 ) is on the directrix, putting this coordinates into the equation of the line gives:

3(-4) + 2(-1) + c = 0  =>  -12 - 2 + c = 0  =>  c = 14.

Therefore the equation of the directrix is

3x + 2y + 14 = 0