JC3187 said:Hi guys,
What is the best way to sketch
0 = x2 - 2x + 4y2 And
8 = x2 - 2x + 4y2
?
How do I sketch these two and how do I know they're both ellipses?
Thank you.
What is the general equation for an ellipse in Cartesian coordinates? Can you see why completing the square helps?JC3187 said:Then what do i do?
The main difference between sketching an ellipse with a 0 and an 8 is the orientation of the ellipse. When sketching an ellipse with a 0, the major axis (longest diameter) is horizontal, while with an 8, the major axis is vertical.
The shape of an ellipse is determined by the eccentricity, which is the ratio of the distance between the foci and the length of the major axis. When comparing 0 and 8, the eccentricity will be the same for both orientations, since it is a measure of the roundness of the ellipse rather than its orientation.
Yes, an ellipse with a 0 orientation can be rotated to become an 8. This is because the orientation of the ellipse is arbitrary and can be changed without altering its shape or eccentricity.
The area of an ellipse remains the same regardless of its orientation. This is because the formula for calculating the area of an ellipse (A = π * a * b) takes into account both the length of the major axis (a) and the minor axis (b), which are the same for both 0 and 8 orientations.
Yes, understanding the differences between 0 and 8 orientations in sketching ellipses is important in fields such as astronomy, engineering, and design. Ellipses are commonly used to represent the orbits of planets, the shape of satellite paths, and the design of objects such as wheels and lenses. Being able to accurately sketch ellipses in different orientations is crucial for these applications.