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1.find the standard equation of a circle with its center at (2, 8) and a radius of 10.
2. find the standard equation of a parabola with its vertex at (2, 2) and a focus at (2, 5).
3. find the standard equation of a parabola with its vertex at (5, 2) and a directrix x =3.

Step-by-step explanation:

1. Equation of a circle:

(x - h)² + (y - k)² = r²

where (h, k) is the center and r is the radius.

If (h, k) is (2, 8) and r = 10:

(x - 2)² + (y - 8)² = 100

2. The vertex and focus have the same x-coordinate, so this is a vertical parabola.  Equation of a vertical parabola is:

y = 1/(4p) (x - h)² + k

where (h, k) is the vertex and p is the distance from the vertex to the focus.

If (h, k) is (2, 2) and p = 5-2 = 3:

y = 1/12 (x - 2)² + 2

3. The directrix is a vertical line, so this is a horizontal parabola.  Equation of a horizontal parabola is:

x = 1/(4p) (y - k)² + h

The distance between the directrix and the vertex is the same as p.

If (h, k) is (5, 2) and p = 5-3 = 2:

x = 1/8 (y - 2)² + 5

g = (2, 2)

step-by-step explanation:

given the ratio is 1 : 1 then f is the midpoint of eg

let the coordinates of g = (x, y)

using the midpoint formula

(0 + x) = 1 ( multiply both sides by 2 )

0 + x = 2 ⇒ x = 2

similarly

(4 + y) = 3 ( multiply both sides by 2 )

4 + y = 6 ( subtract 4 from both sides )

y = 2

coordinates of g = (2, 2)