Question

Rotate the axes to eliminate the xy-term in the equation. Then write the equation in standard form.

61x^2 - 22xy + 61y^2 - 1800 = 0

Answer 1

Suppose we need to rotate the ellipse clockwise by the angle α. Rotating by the angle α moves the point (x,y) to the point (x cos(α) − y sin(α), y cos(α) + x sin(α))

Let's plug this into the equation:

61(x cos(α) − y sin(α))² - 22(x cos(α) − y sin(α))(y cos(α) + x sin(α)) + 61(y cos(α) + x sin(α)) - 1800 = 0

When you simplify this equation using algebra and the dust settles, you should get this:
(61 - 22cos(α)sin(α))x² + (61 + 22cos(α)sin(α))y² - 22(cos²α - sin²α)xy = 1800

Now, the xy term has to be zero in order for the ellipse to be upright. Therefore, cos²α = sin²α. The simplest solution for this is α = 45 degrees.
Plugging this in, we get:

50x² + 72y² = 1800
x²/36 + y²/25 = 1
x² / 6² + y² / 5² = 1

Answer 2

thanks so much!

Answer 3

My pleasure.

Answer 4

Can you help me with this one?

Rotate the axes to eliminate the xy-term in the equation. Then write the equation in standard form.

97x2 + 130xy + 97y2 - 2592 = 0