Review: analytic geometry

Analytic geometry, grade 12 (5 units) Lesson 21 of 21Practice

Review: analytic geometry

The course’s tools

Points and lines: the dividing formula, the formula for a point’s distance from a line (the line in implicit form), the number of solutions of a system – the position.
A locus: $\displaystyle P(x,y)$ , the condition as an equation, simplifying until the shape is recognised.
The circle: completing the square, the tangent by the formula, the position by the distance from the centre or between the centres.
The parabola, the ellipse, the hyperbola: the definition as a locus, the canonical equation, $\displaystyle p$ , $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ ; with a line and with a circle – a system of equations, and a tangent when the discriminant is $\displaystyle 0$ .

$\displaystyle A(1,-2)$ and $\displaystyle B(9,2)$ .
The point $\displaystyle C$ divides $\displaystyle AB$ in the ratio $\displaystyle {AC:CB}={3:1}$ .

What are the coordinates of $\displaystyle C$ ?

The distance of

\displaystyle C

from the line

\displaystyle {3x+4y+5}={0}

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The centre of the circle is $\displaystyle M(3,-2)$ , and the circle passes through $\displaystyle C(7,1)$ .
Write the equation of the circle and the equation of its tangent at $\displaystyle C$ , each in a row of its own.

\displaystyle M=(3,-2)

\displaystyle C=(7,1)

The circles $\displaystyle {x^{2}+y^{2}}={25}$ and $\displaystyle {(x-6)^{2}+(y-8)^{2}}={25}$ .

How do the circles lie? Explain.

The parabola $\displaystyle {y^{2}}={8x}$ and the line $\displaystyle {y}={2x-8}$ .

Find the common points.

Write the tangents to the parabola at the two points.

\displaystyle y^{2}=8x

\displaystyle y=2x-8

An ellipse’s foci are $\displaystyle (\pm4,0)$ , and it passes through $\displaystyle (0,3)$ .
Write the equation of the ellipse.

\displaystyle (-4,0)

\displaystyle (4,0)

\displaystyle (0,3)

A hyperbola’s asymptotes are $\displaystyle {y}={\pm\frac{3}{4}x}$ , and one of its foci is $\displaystyle (5,0)$ .
Write the equation of the hyperbola.

\displaystyle y=\frac{3}{4}x

\displaystyle y=-\frac{3}{4}x

\displaystyle (5,0)

Find the locus of the points as far from $\displaystyle (0,-2)$ as from the line $\displaystyle {y}={2}$ .

\displaystyle y=2

\displaystyle (0,-2)

For which values of $\displaystyle n$ does the line $\displaystyle {y}={x+n}$ touch the ellipse $\displaystyle {\frac{x^{2}}{9}+\frac{y^{2}}{16}}={1}$ ?

Formulas: analytic geometry, grade 12 (5 units)

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Points and lines

The distance between the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{(x_1-x_2)^{2}+(y_1-y_2)^{2}}$
The slope of the line through the two points
$\displaystyle m$ $\displaystyle =$ $\displaystyle \frac{y_2-y_1}{x_2-x_1}$ $\displaystyle x_1\neq x_2$
The line with slope $\displaystyle m$ through $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1=m(x-x_1)$
Perpendicular lines – $\displaystyle m_1$ , $\displaystyle m_2$ their slopes
$\displaystyle m_1\cdot m_2$ $\displaystyle =$ $\displaystyle -1$
The point $\displaystyle C$ dividing the segment $\displaystyle AB$ in the ratio $\displaystyle {AC:CB}={k:l}$ – $\displaystyle A(x_1,y_1)$ , $\displaystyle B(x_2,y_2)$
$\displaystyle C\left(\frac{lx_1+kx_2}{k+l},\frac{ly_1+ky_2}{k+l}\right)$
The distance of the point $\displaystyle (x_0,y_0)$ from the line $\displaystyle {Ax+By+C}={0}$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{|Ax_0+By_0+C|}{\sqrt{A^{2}+B^{2}}}$
The distance between the parallel lines $\displaystyle {Ax+By+C_1}={0}$ and $\displaystyle {Ax+By+C_2}={0}$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{|C_1-C_2|}{\sqrt{A^{2}+B^{2}}}$

Circle

A circle with centre $\displaystyle (a,b)$ and radius $\displaystyle R$
$\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}$
The expanded equation – a circle when, after completing the square, the right side is positive
$\displaystyle x^{2}+y^{2}+Cx+Dy+E=0$
The tangent to the circle at the point $\displaystyle (x_0,y_0)$ on it
$\displaystyle (x_0-a)(x-a)+(y_0-b)(y-b)=R^{2}$

Parabola

The canonical parabola
$\displaystyle y^{2}=2px$
The focus
$\displaystyle F\left(\frac{p}{2},0\right)$
The directrix
$\displaystyle x=-\frac{p}{2}$
The tangent at the point $\displaystyle (x_0,y_0)$ on the parabola
$\displaystyle y\cdot y_0=p(x+x_0)$

Ellipse

The canonical ellipse – the foci on the $\displaystyle x$ -axis when $\displaystyle a>b$
$\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
The distance of a focus from the origin
$\displaystyle c$ $\displaystyle =$ $\displaystyle \sqrt{a^{2}-b^{2}}$
The sum of the distances of a point $\displaystyle P$ on the ellipse from the foci
$\displaystyle PF_1+PF_2$ $\displaystyle =$ $\displaystyle 2a$

Hyperbola

The canonical hyperbola – the foci on the $\displaystyle x$ -axis
$\displaystyle \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
The distance of a focus from the origin
$\displaystyle c$ $\displaystyle =$ $\displaystyle \sqrt{a^{2}+b^{2}}$
The difference of the distances of a point $\displaystyle P$ on the hyperbola from the foci
$\displaystyle |PF_1-PF_2|$ $\displaystyle =$ $\displaystyle 2a$
The asymptotes
$\displaystyle y=\pm\frac{b}{a}x$