Question

Find an equation for the hyperbola with center at (8, 2); focus at (5, 2); vertex at (7, 2)

Update : Find the equation of the parabola with vertex at (1, 7) and focus at (6, 7)

Answer 1

(1)

The Center is at (8,2)  and focus at (5,2) and vertex at (7,2)

The Centre, focus, vertex all lie on  horizontal line y=2.

Distance from Centre (8,2) to vertex (7,2)

Hence the vertex is 1 unit from the Centre that is a=1.

Distance from Centre (8,2) to focus (5,2) :

(8,2)(5,2)

Hence the focus is 3 units from the Centre that is c=3.

From 

Now ,the equation of the hyperbola is .

=>.

Hence the equation of the hyperbola is .

Answer 2

(2)

The Vertex of Parabola (h,k)  is (1,7)

The focus point of parabola is (6,7)

As the y coordinates are Equal then the Equation of parabola is (x - h)² = 4p (y-k)

Distance Between Vertex and Focus is p = 5.

So the Equation of Parabola is

(x - 1)² = 20(y - 7)

Therefore the Equation of Parabola is (x - 1)² = 20(y - 7).