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?

The maximum number of solutions is one.
The maximum number of solutions is three.
The circle has a radius equal to 3.
The circle has a radius less than 9.

Answer 1

"The maximum number of solutions is one."

Explanation:

Hopefully the drawing helps visualize the problem.

The circle has a radius of 9 because the vertex is 9 units above the center of the circle.

The circle the parabola intersect only once and cannot intercept more than once.

The solution is "The maximum number of solutions is one."

Let's see if we can find an algebraic way:

The equation for the circle given as we know from the problem without further analysis is so far
`x^2+y^2=r^2`.

The equation for the parabola without further analysis is
`y=ax^2+9`.

We are going to plug
`ax^2+9` into
`x^2+y^2=r^2` for
`y`.

`x^2+y^2=r^2`

`x^2+(ax^2+9)^2=r^2`

To expand
`(ax^2+9)^2`, I'm going to use the following formula:

`(u+v)^2=u^2+2uv+v^2`.

`(ax^2+9)^2=a^2x^4+18ax^2+81`.

`x^2+y^2=r^2`

`x^2+(ax^2+9)^2=r^2`

`x^2+a^2x^4+18ax^2+81=r^2`

So this is a quadratic in terms of
`x^2`

Let's put everything to one side.

Subtract
`r^2` on both sides.

`x^2+a^2x^4+18ax^2+81-r^2=0`

Reorder in standard form in terms of x:

`a^2x^4+(18a+1)x^2+(81-r^2)=0`

The discriminant of the left hand side will tell us how many solutions we will have to the equation in terms of
`x^2`.

The discriminant is
`B^2-4AC`.

If you compare our equation to
`Au^2+Bu+C`, you should determine
`A=a^2`

`B=(18a+1)`

`C=(81-r^2)`

The discriminant is

`B^2-4AC`

`(18a+1)^2-4(a^2)(81-r^2)`

Multiply the (18a+1)^2 out using the formula I mentioned earlier which was:

`(u+v)^2=u^2+2uv+v^2`

`(324a^2+36a+1)-4a^2(81-r^2)`

Distribute the 4a^2 to the terms in the ( ) next to it:

`324a^2+36a+1-324a^2+4a^2r^2`

`36a+1+4a^2r^2`

We know that
`a>0` because the parabola is open up.

We know that
`r>0` because in order it to be a circle a radius has to exist.

So our discriminat is positive which means we have two solutions for
`x^2`.

But how many do we have for just
`x`.

We have to go further to see.

So the quadratic formula is:

`(-B \pm √(B^2-4AC))/(2A)`

We already have
`B^2-4AC}`

`(-(18a+1) \pm √(36a+1+4a^2r^2))/(2a^2)`

This is t he solution for
`x^2`.

To find
`x` we must square root both sides.

`x=\pm \sqrt{(-(18a+1) \pm √(36a+1+4a^2r^2))/(2a^2)}`

So there is only that one real solution (it actually includes 2 because of the plus or minus outside) here for x since the other one is square root of a negative number.

That is,

`x=\pm \sqrt{(-(18a+1) \pm √(36a+1+4a^2r^2))/(2a^2)}`

means you have:

`x=\pm \sqrt{(-(18a+1)+√(36a+1+4a^2r^2))/(2a^2)}`

or

`x=\pm \sqrt{(-(18a+1)-√(36a+1+4a^2r^2))/(2a^2)}`.

The second one is definitely includes a negative result in the square root.

18a+1 is positive since a is positive so -(18a+1) is negative

2a^2 is positive (a is not 0).

So you have (negative number-positive number)/positive which is a negative since the top is negative and you are dividing by a positive.

We have confirmed are max of one solution algebraically. (It is definitely not 3 solutions.)

If r=9, then there is one solution.

If r>9, then there is two solutions as this shows:

`x=\pm \sqrt{(-(18a+1)+√(36a+1+4a^2r^2))/(2a^2)}`

r=9 since our circle intersects the parabola at (0,9).

Also if (0,9) is intersection, then

`0^2+9^2=r^2` which implies r=9.

Plugging in 9 for r we get:

`x=\pm \sqrt{(-(18a+1)+√(36a+1+4a^2(9)^2))/(2a^2)}`

`x=\pm \sqrt{(-(18a+1)+√(36a+1+324a^2))/(2a^2)}`

`x=\pm \sqrt{(-(18a+1)+√((18a+1)^2))/(2a^2)}`

`x=\pm \sqrt{(-(18a+1)+18a+1)/(2a^2)}`

`x=\pm \sqrt{(0)/(2a^2)}`

`x=\pm 0`

`x=0`

The equations intersect at x=0. Plugging into
`y=ax^2+9` we do get
`y=a(0)^2+9=9`.

After this confirmation it would be interesting to see what happens with assume algebraically the solution should be (0,9).

This means we should have got x=0.

`0=(-(18a+1)+√(36a+1+4a^2r^2))/(2a^2)`

A fraction is only 0 when it's top is 0.

`0=-(18a+1)+√(36a+1+4a^2r^2)`

Add 18a+1 on both sides:

`18a+1=\sqrt{36a+1+4a^2r^2`

Square both sides:

`324a^2+36a+1=36a+1+4a^2r^2`

Subtract 36a and 1 on both sides:

`324a^2=4a^2r^2`

Divide both sides by
`4a^2`:

`81=r^2`

Square root both sides:

`9=r`

The radius is 9 as we stated earlier.

Let's go through the radius choices.

If the radius of the circle with center (0,0) is less than 9 then the circle wouldn't intersect the parabola. So It definitely couldn't be the last two choices.

Answer 2

Option A.

Explanation:

A circle was drawn with the center at origin (0, 0) and a point (0, 9) on the circle.

So the radius will be r =
`\sqrt{(0-0)+(0-9)^(2)}=9`

Equation of this circle will be in the form of x² + y² = r²

Here r represents radius.

So the equation of the circle will be x² + y² = 9²

Or x² + y² = 81

Now we will form the equation of the parabola having vertex at (0, 9)

y² = (x - h)² + k

where (h, k) is the vertex.

Equation of the parabola will be y² = (x - 0)² + 9

y² = x² + 9

Now we will replace the value of y² from this equation in the equation of circle to get the solution of this system of the equations.

x² + x² + 9 = 81

2x² = 81 - 9

2x² = 72

x²= 36

x = ±√36

x = ±6

Since circle and parabola both touch on a single point (0, 9) therefore, there will be only one solution that is x = 6.

For x = 6,

6² + y² = 9²

36 + y² = 81

y² = 81 - 36 = 45

y = √45 = 3√5

Option A. will be the answer.