Find the equation of the ellipse having axes along the coordinate axes and passing through the points (4 ,3), (-1,4).
just plug your points into
x^2/a^2 + y^2/b^2 = 1
then solve for 1/a^2 and 1/b^2
To find the equation of the ellipse, we need to determine its center, semi-major axis, and semi-minor axis.
Step 1: Find the center of the ellipse
Since the ellipse has its axes along the coordinate axes, the center of the ellipse is simply the midpoint between the two given points. To find the midpoint, we average the x-coordinates and the y-coordinates separately:
x-coordinate of center = (4 + (-1)) / 2 = 3/2 = 1.5
y-coordinate of center = (3 + 4) / 2 = 7/2 = 3.5
Therefore, the center of the ellipse is (1.5, 3.5).
Step 2: Find the semi-major axis (a)
The semi-major axis (a) is the distance from the center of the ellipse to one of the points. Since the ellipse passes through the point (4, 3), we can use the distance formula to find the distance between the center and this point:
a = √((x2 - x1)^2 + (y2 - y1)^2) = √((4 - 1.5)^2 + (3 - 3.5)^2) = √(2.5^2 + 0.5^2) = √6.5 ≈ 2.55.
Step 3: Find the semi-minor axis (b)
Since the axes of the ellipse align with the coordinate axes, the semi-minor axis (b) will be the distance from the center to a point on the ellipse that lies on the y-axis. The y-coordinate of this point is 4, so we can use the distance formula again:
b = |4 - 3.5| = 0.5
We now have all the required information to write the equation of the ellipse.
Step 4: Write the equation of the ellipse
The general form of the equation for an ellipse with its center at (h, k) is:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
Plugging in the values we found, the equation of the ellipse is:
(x - 1.5)^2 / (2.55)^2 + (y - 3.5)^2 / (0.5)^2 = 1
So, the equation of the ellipse with axes along the coordinate axes and passing through the points (4, 3) and (-1, 4) is:
(x - 1.5)^2 / 6.502 + (y - 3.5)^2 / 0.252 = 1