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Eliminating the parameter practice problems

Eliminate the parameter to find the curve's equation in x and y, and name it. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

6 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 8.371

medium
Eliminate the parameter from \( \displaystyle x = 2 \sec{\left(t \right)},\ y = 2 \tan{\left(t \right)} \), and say what kind of curve it is.
  1. sec²t − tan²t = 1.
    Reviewed
  2. \[ \left. \frac{x^{2}}{4} - \frac{y^{2}}{4} - 1 \right|_{\substack{ x=2 \sec{\left(t \right)}\\ y=2 \tan{\left(t \right)} }} = 0 \]
    The parametric point satisfies the equation for every t.✓ Proved
Answer \( \frac{x^{2}}{4} - \frac{y^{2}}{4} = 1\ (\text{hyperbola}) \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.232

easy
Eliminate the parameter from \( \displaystyle x = t + 3,\ y = t^{2} - 2 \), and say what kind of curve it is.
  1. t = x − 3.
    Reviewed
  2. \[ \left. - x^{2} + 6 x + y - 7 \right|_{\substack{ x=t + 3\\ y=t^{2} - 2 }} = 0 \]
    The parametric point satisfies the equation for every t.✓ Proved
Answer \( y = x^{2} - 6 x + 7\ (\text{parabola}) \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.347

easy
Eliminate the parameter from \( \displaystyle x = - t - 2,\ y = 3 t + 3 \), and say what kind of curve it is.
  1. t = (x − -2)/-1; substitute into y.
    Reviewed
  2. \[ \left. 3 x + y + 3 \right|_{\substack{ x=- t - 2\\ y=3 t + 3 }} = 0 \]
    The parametric point satisfies the equation for every t.✓ Proved
Answer \( y = - 3 x - 3\ (\text{line}) \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.365

easy
Eliminate the parameter from \( \displaystyle x = - 4 t - 1,\ y = 2 - 4 t \), and say what kind of curve it is.
  1. t = (x − -1)/-4; substitute into y.
    Reviewed
  2. \[ \left. - x + y - 3 \right|_{\substack{ x=- 4 t - 1\\ y=2 - 4 t }} = 0 \]
    The parametric point satisfies the equation for every t.✓ Proved
Answer \( y = x + 3\ (\text{line}) \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.488

easy
Eliminate the parameter from \( \displaystyle x = t + 2,\ y = t^{2} + 3 \), and say what kind of curve it is.
  1. t = x − 2.
    Reviewed
  2. \[ \left. - x^{2} + 4 x + y - 7 \right|_{\substack{ x=t + 2\\ y=t^{2} + 3 }} = 0 \]
    The parametric point satisfies the equation for every t.✓ Proved
Answer \( y = x^{2} - 4 x + 7\ (\text{parabola}) \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.532

easy
Eliminate the parameter from \( \displaystyle x = - 2 t - 2,\ y = 2 t \), and say what kind of curve it is.
  1. t = (x − -2)/-2; substitute into y.
    Reviewed
  2. \[ \left. x + y + 2 \right|_{\substack{ x=- 2 t - 2\\ y=2 t }} = 0 \]
    The parametric point satisfies the equation for every t.✓ Proved
Answer \( y = - x - 2\ (\text{line}) \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More eliminating the parameter practice

Eliminate the parameter from \( \displaystyle x = t + 2,\ y = t^{2} - 2 \), and say what kind of curve it is.
Problem 8.221easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = t,\ y = t^{2} + 3 \), and say what kind of curve it is.
Problem 8.222easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 4 t + 3,\ y = 2 t - 2 \), and say what kind of curve it is.
Problem 8.230easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = - 2 t - 1,\ y = - t - 1 \), and say what kind of curve it is.
Problem 8.234easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = t + 1,\ y = t^{2} + 1 \), and say what kind of curve it is.
Problem 8.338easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = t - 2,\ y = t^{2} + 2 \), and say what kind of curve it is.
Problem 8.339easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 2 - 2 t,\ y = 2 - 4 t \), and say what kind of curve it is.
Problem 8.342easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = t - 2,\ y = t^{2} \), and say what kind of curve it is.
Problem 8.344easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 3 t,\ y = t - 3 \), and say what kind of curve it is.
Problem 8.345easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = t - 1,\ y = t^{2} - 1 \), and say what kind of curve it is.
Problem 8.360easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = - 2 t - 2,\ y = t - 3 \), and say what kind of curve it is.
Problem 8.366easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 3 - t,\ y = 2 t \), and say what kind of curve it is.
Problem 8.368easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 3 t + 1,\ y = - 2 t \), and say what kind of curve it is.
Problem 8.370easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = t - 1,\ y = t^{2} + 3 \), and say what kind of curve it is.
Problem 8.485easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = t + 2,\ y = t^{2} - 1 \), and say what kind of curve it is.
Problem 8.491easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = t + 1,\ y = t^{2} + 3 \), and say what kind of curve it is.
Problem 8.493easy✓ Nihil obstat
Eliminate the parameter from \( \displaystyle x = t - 1,\ y = t^{2} - 2 \), and say what kind of curve it is.
Problem 8.538easy✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 3 \cos{\left(t \right)},\ y = 3 \sin{\left(t \right)} + 1 \), and say what kind of curve it is.
Problem 8.220medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = e^{t},\ y = e^{4 t} \), and say what kind of curve it is.
Problem 8.223medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 3 \sec{\left(t \right)},\ y = 3 \tan{\left(t \right)} \), and say what kind of curve it is.
Problem 8.224medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = \sec{\left(t \right)},\ y = \tan{\left(t \right)} \), and say what kind of curve it is.
Problem 8.225medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 2 \sec{\left(t \right)},\ y = \tan{\left(t \right)} \), and say what kind of curve it is.
Problem 8.226medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 3 \cos{\left(t \right)},\ y = 3 \sin{\left(t \right)} - 2 \), and say what kind of curve it is.
Problem 8.227medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = \cos{\left(t \right)} + 1,\ y = 4 \sin{\left(t \right)} - 3 \), and say what kind of curve it is.
Problem 8.228medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 4 \cos{\left(t \right)} - 3,\ y = 4 \sin{\left(t \right)} + 1 \), and say what kind of curve it is.
Problem 8.229medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 3 \cos{\left(t \right)} - 3,\ y = 4 \sin{\left(t \right)} \), and say what kind of curve it is.
Problem 8.231medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 3 \sec{\left(t \right)},\ y = \tan{\left(t \right)} \), and say what kind of curve it is.
Problem 8.233medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = \cos{\left(t \right)} - 2,\ y = 4 \sin{\left(t \right)} - 3 \), and say what kind of curve it is.
Problem 8.340medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 2 \cos{\left(t \right)} - 1,\ y = 2 \sin{\left(t \right)} - 3 \), and say what kind of curve it is.
Problem 8.341medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 2 \cos{\left(t \right)} + 2,\ y = \sin{\left(t \right)} \), and say what kind of curve it is.
Problem 8.343medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 3 \cos{\left(t \right)} + 1,\ y = 3 \sin{\left(t \right)} + 2 \), and say what kind of curve it is.
Problem 8.346medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 5 \cos{\left(t \right)} - 1,\ y = 5 \sin{\left(t \right)} + 3 \), and say what kind of curve it is.
Problem 8.358medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = e^{t},\ y = e^{5 t} \), and say what kind of curve it is.
Problem 8.359medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = \cos{\left(t \right)} - 1,\ y = \sin{\left(t \right)} + 3 \), and say what kind of curve it is.
Problem 8.361medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = e^{t},\ y = e^{3 t} \), and say what kind of curve it is.
Problem 8.362medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 5 \sec{\left(t \right)},\ y = \tan{\left(t \right)} \), and say what kind of curve it is.
Problem 8.363medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = e^{t},\ y = e^{t} \), and say what kind of curve it is.
Problem 8.364medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 5 \cos{\left(t \right)} + 1,\ y = 4 \sin{\left(t \right)} + 2 \), and say what kind of curve it is.
Problem 8.367medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = \cos{\left(t \right)} + 1,\ y = 4 \sin{\left(t \right)} + 1 \), and say what kind of curve it is.
Problem 8.369medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 5 \cos{\left(t \right)} + 1,\ y = \sin{\left(t \right)} - 1 \), and say what kind of curve it is.
Problem 8.372medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 2 \cos{\left(t \right)},\ y = 2 \sin{\left(t \right)} \), and say what kind of curve it is.
Problem 8.486medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 2 \cos{\left(t \right)} + 2,\ y = 2 \sin{\left(t \right)} - 3 \), and say what kind of curve it is.
Problem 8.487medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 2 \cos{\left(t \right)} + 2,\ y = 2 \sin{\left(t \right)} - 2 \), and say what kind of curve it is.
Problem 8.489medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = \cos{\left(t \right)} + 1,\ y = \sin{\left(t \right)} - 2 \), and say what kind of curve it is.
Problem 8.490medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 5 \cos{\left(t \right)},\ y = 5 \sin{\left(t \right)} + 3 \), and say what kind of curve it is.
Problem 8.492medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 5 \cos{\left(t \right)} - 2,\ y = 4 \sin{\left(t \right)} \), and say what kind of curve it is.
Problem 8.494medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = \sec{\left(t \right)},\ y = 2 \tan{\left(t \right)} \), and say what kind of curve it is.
Problem 8.531medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 2 \sec{\left(t \right)},\ y = 3 \tan{\left(t \right)} \), and say what kind of curve it is.
Problem 8.533medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 5 \sec{\left(t \right)},\ y = 3 \tan{\left(t \right)} \), and say what kind of curve it is.
Problem 8.534medium✓ Nihil obstat
Eliminate the parameter from \( \displaystyle x = 4 \cos{\left(t \right)} + 3,\ y = 4 \sin{\left(t \right)} + 3 \), and say what kind of curve it is.
Problem 8.535medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 2 \cos{\left(t \right)} - 3,\ y = 4 \sin{\left(t \right)} + 2 \), and say what kind of curve it is.
Problem 8.536medium✓ Every equation proved
Eliminate the parameter from \( \displaystyle x = 3 \sec{\left(t \right)},\ y = 4 \tan{\left(t \right)} \), and say what kind of curve it is.
Problem 8.537medium✓ Nihil obstat
Eliminate the parameter from \( \displaystyle x = 4 \sec{\left(t \right)},\ y = 2 \tan{\left(t \right)} \), and say what kind of curve it is.
Problem 8.539medium✓ Nihil obstat
Eliminate the parameter from \( \displaystyle x = \cos{\left(t \right)} - 3,\ y = \sin{\left(t \right)} - 1 \), and say what kind of curve it is.
Problem 8.540medium✓ Every equation proved