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

1 Answer
Andy Y.
Jun 6, 2017

y=sqrt(2)/4(x-1)^2+3y=√24(x−1)2+3

Explanation:

Vertex form of a parabola can be expressed as

y=a(x-h)^2+ky=a(x−h)2+k

or

4p(y-k)=(x-h)^24p(y−k)=(x−h)2

Where 4p=1/a4p=1a is the distance between the vertex and the focus.

The distance formula is

1/a=sqrt((x_2-x_1)^2+(y_2-y_1)^2)1a=√(x2−x1)2+(y2−y1)2

Let's call (x_1,y_1)=(3,5)(x1,y1)=(3,5) and (x_2,y_2)=(1,3)(x2,y2)=(1,3). So,

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

Cross multiplying gives a=1/(2sqrt(2))=sqrt(2)/4a=12√2=√24

The final, vertex form is therefore,

y=sqrt(2)/4(x-1)^2+3y=√24(x−1)2+3

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