what is the equation of an ellipse that has an center at (4,-3) and passes through (1,-3) and (4,2) ?
The equation of an ellipse is: x^2/a^2 + y^2/b^2 = 1,
where a and b are the lengths of the semi-axes.
If the ellipse is centered at (h,k), then its equation becomes,
(x-h)^2/a^2 + (y-k)^2/b^2 = 1.
Since, in our case, we have (h,k) = (4,-3), then our ellipse is,
(x-4)^2/a^2 + (y+3)^2/b^2 = 1.
Now use the coordinates of the two points lying on the ellipse to give two equations in two unknowns, viz. a and b, and solve for them.
Point = (1, -3)
(1-4)^2/a^2 + (-3+3)^2/b^2 = 1.
9/a^2 = 1
a = 3 (a is the length of a semi-axis, so only takes positive values)
Point = (4,2)
(4-4)^2/a^2 + (2+3)^2/b^2 = 1
25/b^2 = 1
b = 5 (b is the length of a semi-axis, so only takes positive values)
The equation of the ellipse is now given by,
(x-4)^2/3^2 + (y+3)^2/5^2 = 1.