What is the vertex form of the parabola with a focus at (3,5) and vertex at (1,3)?

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Answer 1 · 2017-06-06T10:37:09

#y=sqrt(2)/4(x-1)^2+3#

Vertex form of a parabola can be expressed as

#y=a(x-h)^2+k#

or

#4p(y-k)=(x-h)^2#

Where #4p=1/a# is the distance between the vertex and the focus.

#1/a=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

Let's call #(x_1,y_1)=(3,5)# and #(x_2,y_2)=(1,3)#. So,

#1/a=sqrt((1-3)^2+(3-5)^2)=sqrt((-2)^2+(-2)^2)=2sqrt(2)#

Cross multiplying gives #a=1/(2sqrt(2))=sqrt(2)/4#

The final, vertex form is therefore,

#y=sqrt(2)/4(x-1)^2+3#