Final answer:
The foci of the given ellipse with the equation x²+y²-4x+9x=23 are located at (2,0).
Step-by-step explanation:
An ellipse is a closed curve where the sum of the distances from any point on the curve to the two foci is constant. To find the foci of the given ellipse with the equation x²+y²-4x+9x=23, we need to rewrite the equation in the standard form: (x-h)²/a² + (y-k)²/b² = 1, where (h,k) represents the coordinates of the center and 'a' and 'b' are the semi-major and semi-minor axes respectively.
In this case, we can rewrite the equation as x²-4x + y²=23-9.
Completing the square, we get (x-2)² + y² = 12.
Comparing with the standard form, we have (x-2)²/12 + y²/12 = 1.
From this equation, we can determine that the center of the ellipse is (2,0), the semi-major axis is √12, and the semi-minor axis is also √12.
Since the foci of an ellipse are located along the major axis at a distance 'c' from the center, where 'c' is given by the equation c² = a² - b², we can calculate the value of 'c' to find the foci.
In this case, c² = 12 - 12 = 0, so 'c' must be 0.
Therefore, the foci of the given ellipse are located at (2,0).