Find the exact solution(s) of the system: (x^2/4)-y^2=1 and x=y^2+1

Answer:(4,sqrt3),(4,-sqrt3)

2)Write an equation for an ellipse if the endpoints of the major axis are at (-8,1) and (8,1) and the endpoints of the minor axis are at (0,-1) and (0,3).

Answer: (x^2/64) + (y-1)^2/4 = 1

Thanks a lot!!

To find the exact solution(s) of a system of equations, you can use substitution or elimination. Let's explain how to solve the first system of equations using substitution method:

1) (x^2/4) - y^2 = 1
2) x = y^2 + 1

We can solve equation 2 for x and substitute it into equation 1:

Substituting x from equation 2 into equation 1:

((y^2 + 1)^2 / 4) - y^2 = 1
(y^4 + 2y^2 + 1)/4 - y^2 = 1

Now, multiply both sides of the equation by 4 to eliminate the fraction:

y^4 + 2y^2 + 1 - 4y^2 = 4

Combine like terms:

y^4 - 2y^2 - 3 = 0

This is a quadratic equation in terms of y^2. Let's solve it by factoring:

(y^2 - 3)(y^2 + 1) = 0

Setting each factor equal to zero:

y^2 - 3 = 0, or y^2 + 1 = 0

Solving for y in each equation:

For y^2 - 3 = 0:
y^2 = 3
y = sqrt(3) or y = -sqrt(3)

For y^2 + 1 = 0:
y^2 = -1
This equation has no real solutions.

Now that we have the values of y, we can substitute them back into equation 2 to find the corresponding x-values:

For y = sqrt(3):
x = (sqrt(3))^2 + 1 = 3 + 1 = 4
So one solution is (4, sqrt(3))

For y = -sqrt(3):
x = (-sqrt(3))^2 + 1 = 3 + 1 = 4
So the other solution is (4, -sqrt(3))

Therefore, the exact solutions of the given system of equations are (4, sqrt(3)) and (4, -sqrt(3)).

Regarding the second question:

To write an equation for an ellipse, we need to determine the coordinates of the center and the lengths of the major and minor axes.

From the given information, the center of the ellipse is (h,k) = (0,1). The major axis endpoints are (-8,1) and (8,1), so the length of the major axis (2a) is equal to 8 - (-8) = 16. Therefore, a = 16/2 = 8. The minor axis endpoints are (0,-1) and (0,3), so the length of the minor axis (2b) is equal to 3 - (-1) = 4. Therefore, b = 4/2 = 2.

Using these values, we can write the equation of the ellipse in standard form:

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1

Substituting the values we obtained:

(x - 0)^2 / 8^2 + (y - 1)^2 / 2^2 = 1

Simplifying the equation gives us:

x^2 / 64 + (y - 1)^2 / 4 = 1

Therefore, the equation of the ellipse is (x^2/64) + ((y-1)^2/4) = 1.

I hope this explanation helps! Let me know if you have any further questions.