Question

Eliminate the parameter to find a cartesian equation of the curve calculator.

A) x = cos(t), y = sin(t)
B) x = 2t, y = 3t²
C) x = t², y = 2t
D) x = eᵗ, y = e(²ᵗ)

Answer 1

To eliminate the parameter and find a cartesian equation of the curve, we need to express x and y solely in terms of each other. Let's go through each option to find the cartesian equation.

Step-by-step explanation:

To eliminate the parameter and find a cartesian equation of the curve, we need to express x and y solely in terms of each other. Let's go through each option to find the cartesian equation:

A) x = cos(t), y = sin(t): We can rewrite x as x^2 + y^2 = cos^2(t) + sin^2(t) = 1. So, the cartesian equation is x^2 + y^2 = 1.

B) x = 2t, y = 3t²: Substituting x = 2t into y = 3t², we get y = 3(2t)² = 12t^2. So, the cartesian equation is y = 12x^2.

C) x = t², y = 2t: Substituting x = t² into y = 2t, we get y = 2(t²) = 2x. So, the cartesian equation is y = 2x.

D) x = e^t, y = e^(2t): Taking the natural logarithm on both sides of x = e^t, we have ln(x) = ln(e^t) = t. Similarly, ln(y) = ln(e^(2t)) = 2t. So, the cartesian equation is ln(y) = 2ln(x), or equivalently, y = e^2x.

Therefore, the cartesian equations of the curves are:

A) x^2 + y^2 = 1

B) y = 12x^2

C) y = 2x

D) y = e^2x