Eliminate the parameter and obtain the standard form of the rectangular equation. hyperbola: x = h + b tan(θ), y = k + a sec(θ) use your result to find a set of parametric equations for the line or conic. (when 0 ≤ θ ≤ 2π. set your center at the origin. enter your answers as a comma-separated list of equations.) hyperbola: vertices: (0, ±2); foci: (0, ± 5 )

Problem 1:

Problem 2:

Step-by-step explanation:

I'm going to make a guess here.

That this is two problems.

Problem 1:

Eliminate the parameter in

and

.

Recall the following Pythagorean Identity:

If we solve our first equation: for the tangent piece we can substitute into this identity.  We will also need to solve for the secant piece in .

Let's do the equation for x first:

Subtract h on both sides:

Divide both sides by b:

.

Let's now look at the equation for y:

Subtract k on both sides:

Divide boht sides by a:

.

Now we are going to substitute this into our identity:

We are going to subtract the tangent squared piece on both sides to get 1 by itself:

This an equation for an hyperbola with center .

The vertices are at .

The foci is at where .

Problem 2:

We want the equation parametric equations for the conic given:

Hyperbola with vertices and foci and center at (0,0).

We are using the information from the first sentence.

We have (h,k)=(0,0).

We have a=2 and c=5 so we need to find b.

Subtract 4 on both sides:

Square root both sides:

Let's put into our formula for the parametric equations now:

.

We don't really need that 0 in there:

5x+40

5(x + 8)

7x - 49

7(x - 7)

18x + 9

9(2x + 1)

15x - 10

5(3x - 2)

step-by-step explanation:

for number one all you have to do is factor 5 out of 5x + 40

for number 2 factor 7 out of 7x - 49

for number three factor 9 out of 18x + 9

for number four factor 5 out of 15x - 10