曲線的參數方程：練習題

A curve is defined by the parametric equations \(x = t^2 + 1\) and \(y = \frac{1}{t}\) for \(t \neq 0\). Which of the following is the Cartesian equation of the curve?

A curve is defined by the parametric equations \(x = t^2 - 4\) and \(y = 2t\) for \(t \in \mathbb{R}\). The line \(y = x - 4\) intersects the curve at two points. Find the distance between these two intersection points.

A curve is defined by the parametric equations \(x = \frac{1}{t+1}\) and \(y = t^2\), where \(t \neq -1\). Express the Cartesian equation of the curve in the form \(y = f(x)\).

A curve is defined by the parametric equations \(x = 2t^2 - 1\) and \(y = 4t\).
Show that the Cartesian equation of the curve is \(x = \frac{y^2}{8} - 1\) and describe the shape of the curve.

A curve is defined by the parametric equations
\(x = 2\cos\theta\),
\(y = 3\sin\theta\),
for \(0 \le \theta < 2\pi\).

(a) Find the Cartesian equation of the curve in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), stating the values of the constants \(a\) and \(b\).
(b) Find the coordinates of the points where the curve intersects the line \(y = \frac{3}{2}x\).

A curve is defined by the parametric equations \(x = 2\cos t\) and \(y = 3\sin t\), where \(0 \le t < 2\pi\).
(a) Show that the Cartesian equation of the curve can be written in the form \(9x^2 + 4y^2 = k\), and state the value of the constant \(k\).
(b) Find the gradient of the curve at the point where \(t = \frac{\pi}{4}\).
(c) Find the equation of the tangent to the curve at the point where \(t = \frac{\pi}{4}\). Give your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants to be determined.
(d) The tangent found in part (c) intersects the \(x\)-axis at point \(A\) and the \(y\)-axis at point \(B\). Find the exact area of the triangle \(OAB\), where \(O\) is the origin.