Conic Sections — practice questions

Find the centre and radius of the circle $x^2 + y^2 + 8x - 6y - 11 = 0$, and state whether the point $(1, 2)$ lies inside, on, or outside it.

A circular irrigation pivot waters a field modelled by $(x - 6)^2 + (y - 4)^2 = 100$, in metres. A straight drainage channel runs along $3x + 4y = 20$. Determine whether the channel passes through the watered area, and find the length of channel inside it.

Find the equation of the circle passing through the three points $A(1, 1)$, $B(7, 3)$ and $C(4, -4)$, and prove that no circle passes through three collinear points.

For the parabola $(y - 2)^2 = -8(x + 1)$, state the vertex, the focus, the directrix, and the direction in which it opens.

A suspension footbridge has a parabolic cable. The two towers are 60 m apart and the cable is attached 12 m above the deck on each tower, sagging to 2 m above the deck at its lowest point. Find the equation of the cable, and the height of the cable 10 m from a tower.

Prove, from the focus–directrix definition, that the set of points equidistant from the point $(0, p)$ and the line $y = -p$ has equation $x^2 = 4py$. Then use the result to show that the latus rectum has length $|4p|$.

For the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1$, state the centre, the lengths of both axes, the coordinates of the foci, and the eccentricity.

Write $4x^2 + y^2 + 24x - 6y + 41 = 0$ in standard form. State the centre, the axes, and the foci, and verify your answer by substituting one vertex.

A satellite orbits Earth in an ellipse with the centre of the Earth at one focus. Its lowest altitude (perigee) is 500 km above the surface and its highest (apogee) is 39,000 km above the surface. Taking Earth's radius as 6,370 km, find the equation of the orbit relative to the ellipse's centre, its eccentricity, and the satellite's distance from Earth's centre when it is at the end of the minor axis.

For the hyperbola $\dfrac{y^2}{9} - \dfrac{x^2}{16} = 1$, state the centre, the vertices, the foci, the equations of the asymptotes, and the direction in which it opens.

Write $9x^2 - 4y^2 + 36x + 8y - 4 = 0$ in standard form. State its centre, vertices, foci and asymptotes, and sketch it using the box method, explaining what the box represents.

Prove that for the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ the branches approach the lines $y = \pm\dfrac{b}{a}x$ but never meet them, and use the result to explain why the rectangular hyperbola $xy = k$ has the coordinate axes as its asymptotes.

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 $t$.