Math C

Conic Sections and Polar Coordinates: Practice Problems

Learn the standard forms of parabolas, ellipses and hyperbolas and their foci, then practise tangents, parametric curves and polar coordinates.

Basic, Standard: Grade 12 · Term 1 / Advanced: Grade 12 · Exam prep

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Parabolas

A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).

Formulas

y^2 = 4px:\ \text{focus } (p,\ 0),\ \text{directrix } x = -p\qquad x^2 = 4py:\ \text{focus } (0,\ p),\ \text{directrix } y = -p

Ellipses and hyperbolas

Ellipse

\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\ (a > b > 0)

Foci $(\pm\sqrt{a^2 - b^2},\ 0)$ , major axis $2a$ , minor axis $2b$ ; the distances to the foci add up to $2a$ .

Hyperbola

\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1

Vertices $(\pm a,\ 0)$ , foci $(\pm\sqrt{a^2 + b^2},\ 0)$ , asymptotes $y = \pm\dfrac{b}{a}x$ ; the distances to the foci differ by $2a$ .

Tangent lines and translations

Tangent lines at $(x_1,\ y_1)$ on the curve:

\frac{x_1x}{a^2} + \frac{y_1y}{b^2} = 1,\qquad \frac{x_1x}{a^2} - \frac{y_1y}{b^2} = 1,\qquad y_1y = 2p(x + x_1)

For tangents with a given slope or for counting intersections with a line, substitute the line and use the discriminant. Equations like $4x^2 + 9y^2 - 8x + 36y + 4 = 0$ become translated standard forms after completing the square.

Parametric equations and polar coordinates

The ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ can be written as $x = a\cos\theta$ , $y = b\sin\theta$ ; eliminate the parameter with $\cos^2\theta + \sin^2\theta = 1$ .

Polar and rectangular coordinates

x = r\cos\theta,\qquad y = r\sin\theta,\qquad r^2 = x^2 + y^2

For example, $r = 4\cos\theta$ becomes $x^2 + y^2 = 4x$ after multiplying by $r$ : a circle with center $(2,\ 0)$ and radius $2$ .

Q1BasicGr 12 · Term 1Conic sections, parametric curves and polar coordinates

Find the hyperbola with foci $(7,\ 0)$ and $(-7,\ 0)$ for which the difference of the distances to the foci is $10$ .

Answer

\frac{x^2}{25} - \frac{y^2}{24} = 1

Solution

Let the hyperbola be $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\ (a > 0,\ b > 0)$ . From the distance difference,
$2a = 10 \quad\Longrightarrow\quad a = 5$
From the foci,
$a^2 + b^2 = 49 \quad\Longrightarrow\quad b^2 = 24$
Therefore
$\frac{x^2}{25} - \frac{y^2}{24} = 1$

Q2StandardGr 12 · Term 1Conic sections, parametric curves and polar coordinates

Among the lines with slope $-2$ tangent to the ellipse $\frac{x^2}{2} + y^2 = 1$ , find the one with positive $y$ -intercept.

Answer

y = -2x + 3

Solution

Substituting $y = -2x + k$ gives a quadratic in $x$ .
$9x^2 - 8kx + 2(k^2 - 1) = 0$
Tangency (discriminant $D = 0$ ) gives
$k^2 = 2 \cdot (-2)^2 + 1 = 9 \quad\Longrightarrow\quad k = \pm 3$
Since the intercept is positive,
$y = -2x + 3$

Q3AdvancedExam prepConic sections, parametric curves and polar coordinates

The ellipse $\frac{x^2}{7} + \frac{y^2}{2} = 1$ and the line $y = -x - 1$ meet at two points. Find the midpoint of the segment joining them.

Answer

\left(-\frac{7}{9},\ -\frac{2}{9}\right)

Solution

Eliminating $y$ ,
$9x^2 + 14x - 7 = 0$
If the roots are $\alpha, \beta$ , the midpoint has $x$ -coordinate
$\frac{\alpha + \beta}{2} = -\frac{7}{9}$
Substituting into the line,
$y = -\frac{2}{9},\qquad \left(-\frac{7}{9},\ -\frac{2}{9}\right)$

Q1BasicGr 12 · Term 1Conic sections, parametric curves and polar coordinates

Find the hyperbola with foci $(7,\ 0)$ and $(-7,\ 0)$ for which the difference of the distances to the foci is $8$ .

Q2StandardGr 12 · Term 1Conic sections, parametric curves and polar coordinates

Find the tangent line to the hyperbola $\frac{x^2}{7} - \frac{y^2}{14} = 1$ at the point $(-3,\ 2)$ .

Q3AdvancedExam prepConic sections, parametric curves and polar coordinates

Find the locus of the points $\mathrm{P}$ whose distance from $\mathrm{F}(5,\ 0)$ and distance from the line $x = \frac{16}{5}$ are in the ratio $5 : 4$ .

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Conic Sections and Polar Coordinates: Practice Problems

Parabolas

Ellipses and hyperbolas

Tangent lines and translations

Parametric equations and polar coordinates

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