Question

Use focal property of hyperbola to construct hyperbola​

Answer 1

Even though this shape seems much harder to conceive of than an ellipse, the hyperbola has a defining focal property that is as simple as the ellipse's. Remember, an ellipse has two foci and the shape can be defined as the set of points in a plane whose distances to these two foci have a fixed sum.

Answer 2

• Even though this form appears to be considerably more difficult to imagine than an ellipse, the hyperbola has a distinguishing focus attribute that is as basic as the ellipses.
• Remember that an ellipse has two foci and that the form is defined as the set of points on a plane whose distances to these two foci have a fixed sum.

• Hyperbolas contain two foci as well, and they may be described as the set of points in a plane whose distances from these two points differ by the same amount.
• So, in the diagram below, |d₂-d₁| = c for any constant c for every point P on the hyperbola.

The usual shape for an upwards and downwards opening hyperbola with foci on the y-axis is:

$\frac{y^2}{a^2} - \frac{x^2}{b^2} =1$

By switching x and y, we get hyperbolas that open to the right and left and have foci on the x-axis.

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

We have the shifting equations for a hyperbola centered on the point (h, k):

For a hyperbola with up and down openings;

$\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2} =1$

For a hyperbola with openings to the left and right;

$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2} =1$