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 $g^2+f^2-c>0$).

3. Parabola

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

• $y^2=4ax$: 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 $2a$. ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$ ($a>b$): vertices $(\pm a,0)$, foci $(\pm c,0)$ with $c^2=a^2-b^2$, major axis $2a$, minor axis $2b$, eccentricity $e={\displaystyle \frac{c}{a}}<1$, latus rectum ${\displaystyle \frac{2b^2}{a}}$. With the major axis on the $y$-axis, swap the roles: ${\displaystyle \frac{x^2}{b^2}}+{\displaystyle \frac{y^2}{a^2}}=1$.

5. Hyperbola

The set of points whose distances from two foci differ by a constant $2a$. ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1$: vertices $(\pm a,0)$, foci $(\pm c,0)$ with $c^2=a^2+b^2$, transverse axis $2a$, conjugate axis $2b$, $e={\displaystyle \frac{c}{a}}>1$, latus rectum ${\displaystyle \frac{2b^2}{a}}$. For ${\displaystyle \frac{y^2}{a^2}}-{\displaystyle \frac{x^2}{b^2}}=1$ 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 ${\displaystyle \frac{x^2}{16}}-{\displaystyle \frac{y^2}{9}}=1$.

$c^2=16+9=25$, $c=5$: $e={\displaystyle \frac{5}{4}}$.

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.