Conic Sections Class 11: notes and important questions

Conic Sections: revision notes

1. Sections of a cone

Cutting a double cone with a plane gives a circle (plane perpendicular to the axis), an ellipse, a parabola (plane parallel to a generator) or a hyperbola (plane meeting both nappes). Through the vertex, the section degenerates to a point, a line or a pair of lines.

2. Circle

Centre \((h, k)\), radius \(r\): \((x - h)^2 + (y - k)^2 = r^2\). The general form \(x^2 + y^2 + 2gx + 2fy + c = 0\) has centre \((-g, -f)\) and radius \(\sqrt{g^2 + f^2 - c}\) (a real circle needs \(g^2 + f^2 - c \gt 0\)).

3. Parabola

The set of points equidistant from a fixed point (focus) and a fixed line (directrix). For \(a \gt 0\):

• \(y^2 = 4ax\): focus \((a, 0)\), directrix \(x = -a\), axis \(y = 0\), opens right; \(y^2 = -4ax\) opens left.
• \(x^2 = 4ay\): focus \((0, a)\), directrix \(y = -a\), opens up; \(x^2 = -4ay\) opens down.
• Latus rectum (focal chord perpendicular to the axis): length \(4a\).

4. Ellipse

The set of points whose distances from two foci add to a constant \(2a\). \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) (\(a \gt b\)): vertices \((\pm a, 0)\), foci \((\pm c, 0)\) with \(c^2 = a^2 - b^2\), major axis \(2a\), minor axis \(2b\), eccentricity \(e = \dfrac ca \lt 1\), latus rectum \(\dfrac{2b^2}{a}\). With the major axis on the \(y\)-axis, swap the roles: \(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\).

5. Hyperbola

The set of points whose distances from two foci differ by a constant \(2a\). \(\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\): vertices \((\pm a, 0)\), foci \((\pm c, 0)\) with \(c^2 = a^2 + b^2\), transverse axis \(2a\), conjugate axis \(2b\), \(e = \dfrac ca \gt 1\), latus rectum \(\dfrac{2b^2}{a}\). For \(\dfrac{y^2}{a^2} - \dfrac{x^2}{b^2} = 1\) the transverse axis is the \(y\)-axis.

Worked example 1

Find the centre and radius of \(x^2 + y^2 + 8x - 2y - 8 = 0\).

\((x + 4)^2 + (y - 1)^2 = 25\): centre \((-4, 1)\), radius \(5\).

Worked example 2

Find the focus and directrix of \(x^2 = 20y\).

\(4a = 20\), \(a = 5\): focus \((0, 5)\), directrix \(y = -5\).

Worked example 3

Find the eccentricity of \(\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1\).

\(c^2 = 16 + 9 = 25\), \(c = 5\): \(e = \dfrac54\).

Common errors

• \(c^2 = a^2 + b^2\) for an ellipse, or \(c^2 = a^2 - b^2\) for a hyperbola: they are the other way round.
• Reading \(a\) from \(4a\): in \(y^2 = 12x\), \(a = 3\), not \(12\).
• Placing the foci on the wrong axis: they lie on the axis of the larger denominator (ellipse) or of the positive term (hyperbola).
• Radius from the general form without the square root, or with the sign of \(c\) wrong.

Exam tips

• First put the equation in standard form (divide so the right side is \(1\)), then read off \(a\) and \(b\).
• Draw a quick sketch to check which axis the foci and vertices lie on.