QuestionDetermine the general equation of the hyperbola from $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ .

Studdy Solution

STEP 1

Assumptions1. The given equation is of a hyperbola. . The general equation of a hyperbola is $\frac{(x-h)^{}}{a^{}}-\frac{(y-k)^{}}{b^{}}=1$ for a hyperbola that opens left and right, and $\frac{(y-k)^{}}{a^{}}-\frac{(x-h)^{}}{b^{}}=1$ for a hyperbola that opens up and down, where (h,k) is the center of the hyperbola, and a and b are the semi-major and semi-minor axes respectively.

STEP 2

The given equation is $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ . We can see that it is in the form of the general equation of a hyperbola that opens left and right.

STEP 3

We will now identify the values of h, k, a and b from the given equation.In the given equation, there are no terms involving (x-h) or (y-k), so we can conclude that the center of the hyperbola is at the origin (h,k) = (0,0).

STEP 4

Next, we identify the values of a and b. These are given by the square roots of the denominators of the fractions in the equation.
$a = \sqrt{16} =4$ $b = \sqrt{9} =3$

STEP 5

Now we have all the values we need to write the general equation of the hyperbola.The general equation of the hyperbola is $\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1$ .

STEP 6

Substitute the values of h, k, a and b into the general equation.
$\frac{(x-0)^{2}}{4^{2}}-\frac{(y-0)^{2}}{3^{2}}=1$

STEP 7

implify the equation.
$\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ So, the general equation of the hyperbola given by $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ is $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ .

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