Parametric Forms & Tangents — practice questions

A curve is given parametrically by $x = 2 + 5\cos\theta$, $y = -1 + 5\sin\theta$. Find its Cartesian equation and describe the curve fully.

A curve has parametric equations $x = 3\sec\theta$, $y = 2\tan\theta$. Find its Cartesian equation, state the coordinates of its vertices and asymptotes, and explain which values of $\theta$ must be excluded and what that means for the curve.

The point $P$ has parameter $t$ on the parabola $x = at^2$, $y = 2at$. Prove that the chord joining the points with parameters $p$ and $q$ has gradient $\dfrac{2}{p+q}$, and deduce the gradient of the tangent at $P$ and the condition on $p$ and $q$ for the chord to pass through the focus.

Find the equations of the tangent and the normal to the circle $x^2 + y^2 = 169$ at the point $(5, 12)$.

Determine the values of $k$ for which the line $y = x + k$ is a tangent to the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$, and find the point of contact for the positive value of $k$.

Prove that the tangent to the rectangular hyperbola $xy = c^2$ at the point $\left(ct,\ \dfrac{c}{t}\right)$ cuts the axes at points $A$ and $B$ such that the point of contact is the midpoint of $AB$, and that the triangle $OAB$ has the same area for every $t$.