what is the equation of an ellipse that has an center at (4,-3) and passes through (1,-3) and (4,2)

Question

Most of my questions are dealing with elipse equations and writing them in standard form.

Answer 1

Standard equation of ellipse is (x-h)^2/a^2+(y-k)^2/b^2 = 1

Center of ellipse (h,k) is given = (4,-3)

(x-4)^2/a^2+(y+3)^2/b^2 = 1

And (1,-3) (4,2) points are passes through the above ellipse.

Now substitute the each point in ellipse equation.

x = 1, y = -3

(1-4)^2/a^2+(-3+3)^2/b^2 = 1

9/a^2 = 1

Cross multiplication.

a^2 = 9

a = 3

x = 4, y = 2

(4-4)^2/a^2+(2+3)^2/b^2 = 1

25/b^2 = 1

Cross multplication.

b^2 = 25

b = 5

Now the equation of ellipse is (x-4)^2/3^2+(y+3)^2/5^2 = 1