Question 1010584


       

Write an Equation
 given focus ({{{1}}}, {{{-3}}}) 
directrix {{{y= -5}}}

One way is try the definition of a parabola and the use of the Distance Formula.  
Cutting past all the instructive description, the directrix is all points, ({{{x}}},{{{-5}}}).  The parabola is a variable set of points, ({{{x}}},{{{y}}}):

so, we use points ({{{1}}}, {{{-3}}}) and ({{{x}}},{{{-5}}})


{{{sqrt((x-1)^2+(y-(-3))^2)=sqrt((x-x)^2+(y-(-5))^2)}}}

{{{sqrt((x-1)^2+(y+3)^2)=sqrt((x-x)^2+(y+5)^2)}}}

{{{(x-1)^2+(y+3)^2=(0)^2+(y+5)^2}}}

{{{x^2-2x+1+y^2+6y+9=y^2+10y+25}}}

{{{x^2-2x+1+9-25=10y-6y}}}

{{{x^2-2x-15=4y}}}

{{{y=(1/4)x^2-(1/2)x-15/4}}}

or
{{{y = (1/4 )(x^2-2x-15)}}}