Question

The equation of the circle passing through the focus of the ellipse x²/16​+y²/9​=1 having the center at (0, 3) is:

(a) x²+(y−3)²=1
(b) x²+(y+3)²=1
(c) x²+(y−3)²=25
(d) x²+(y+3)²=25

Answer 1

The equation of the circle passing through the focus of the given ellipse is x² + (y + 3)² = 1.

Step-by-step explanation:

The equation of the circle passing through the focus of the ellipse x²/16 + y²/9 = 1 having the center at (0, 3) is x² + (y + 3)² = 1.

To find the equation of the circle passing through the focus of the ellipse, we need to determine the coordinates of the focus. The center of the ellipse is (0, 3) and the equation of the ellipse is given by x²/16 + y²/9 = 1. Rearranging the equation, we have x²/16 = 1 - y²/9.

Solving for x and taking the square root, we get x = ±4√(1 - y²/9). Since the focus lies on the y-axis, we can choose either positive or negative square root. The focus coordinates are (±4√(1 - y²/9), 0).

Substituting the focus coordinates into the equation of a circle, we get (x - ±4√(1 - y²/9))² + (y - 0)² = r². Simplifying the equation, we have (x - ±4√(1 - y²/9))² + y² = r².

Since the focus is on the y-axis, we can simplify further to get x² + (y ± 4)² = r². Since the center of the circle is (0, 3), we can substitute these values into the equation to get x² + (y + 3)² = r².

Since the equation is in standard form, we can conclude that the equation of the circle passing through the focus of the ellipse is x² + (y + 3)² = 1.