Question

The center of a circle is at the origin on a coordinate grid. The vertex of a parabola that opens upward is at (0.9). If the

circle intersects the parabola at the parabola's vertex, which statement must be true?

Answer 1

The circle has radius = 9 units, and its equation is:

`x^2+y^2=81`

The parabola has equation of the form:

`y=a\,x^2+9`

with
`a` being a positive number

Explanation:

Given the characteristics of the two figures (circle and parabola) as described,

since the parabola intersects the circle at a single point (it vertex) then the radius of the circle centered at the origin must be 9. that makes the equation of the circle to have the following standard form;

`x^2+y^2=9^2\\x^2+y^2=81`

Since there is no other info regarding the parabola, the most one can say about the equation of the parabola, is that in vertex form, the parabola's equation must be of the form:

`y=a\,(x-x_(vertex))^2+y_(vertex)\\y=a\,(x-0)^2+9\\y=a\,x^2+9`

with
`a` being a positive number.