Question

Find the intercepts and domain and perform the symmetry test on each of the following hyperbol (a) 9x?- 16y? = 144 (c) 25x2 - 4y? = 100 (g) -9x2 + 16y= 144 (e) x² + y² = 1 (b) 9x² - y² = 9 (d) 25x - 36 = 900 (0) -25x + 4y = 100 (h) -x + 4y = 16 mantiene foci endnoints of the conine fundamental rectangle an

Answer 1

We are asked to find the intercepts and symmetry test on the following hyperbola:

- x^2 + y^2 = 1

Recall the formulas for vertical transverse axis hyperbolas given in the following image:

This type of hyperbolas have a POSITIVE term in ÿ", and a NEGATIVE term in x.

The image represents the type of graph you get with a hyperbola of this type. And it gives you how to find intercepts if any on each coordinate axis.

As we can see, the a and b parameters are 1 for both in our case, since y^2 is not accompanied by any factor but "1" (one), and the same for the x variable.

Therefore: We don't expect any x intercept (the branches of the hyperbola would NOT cross the x axis.

We expect y-intercepts at the values y=1 and y = -1.

The Domain is all the real axis. The graph of one branch is symmetric to the other branch around the x-axis.

The vertices of the hyperbolas branches are : (0, 1) and (0, -1)

The foci are located on the vertical y axis at the values given by :

`\begin{gathered} (0,\sqrt[]{1^2+1^2})=(0,\sqrt[]{2}) \\ \text{and} \\ \\ (0,-\sqrt[]{1^2+1^2})=(0,-\sqrt[]{2}) \end{gathered}`

And here the actual graph of the given conic equation is shown below:

Notice that for the hyperbolas that follow problem e) seem to be of this same type (where the x term in the hyperbola's equation is NEGATIVE, and the y term is POSITIVE)