# Write the equation of the hyperbola if the short axis is 2*sqrt20 and the point M(10,-sqrt5) is on the hyperbola.

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The equation for the hyperbola is:

x^2/a^2 - y^2/b^2 = 1

let b be the short semi- axis

==> b = 2sqrt20/2 = sqrt20

Now substitute in the equation:

==> x^2/a^2 - y^2/20 = 1

M(10.-sqrt5) is on the hyperbola:

Then we can substitute with x and y values in order to obtain a:

==> 100/a^2 - 5/20 = 1

==> 100/a^2 = 1+ 1/4

==> 100/a^2 = 5/4

==> a^2 = 100*4/5 = 20*4 = 80

==> a = sqrt80 =

Then the hyperbola is:

x^2/80 + y^2/20 = 1

The equation of the hyperbola is:

x^2/a^2 - y^2/b^2 = 1

where a = long semi-axis

b = short semi-axis

To write the equation of the hyperbola we'll have to calculate a, because b is given by the enunciation.

We also know that the hyperbola passes through the point M, meaning that the coordinates of M are verifying the equation of the hyperbola.

xM^2/a^2 - yM^2/b^2 = 1

M(10,-sqrt5) belongs to hyperbola if and only if:

10^2/a^2 - (-sqrt5)^2/20 = 1

100/a^2 - 5/20 = 1

We'll add 5/20 both sides:

100/a^2 = 1 + 5/20

100/a^2 = 25/20

We'll divide by 25 both sides:

4/a^2 = 1/20

a^2 = 4*20

Now, we'll write the equation of the hyperbola:

x^2/a^2 - y^2/b^2 = 1

**x^2/80 - y^2/20 = 1**