Practice: points and lines

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

Practice: points and lines

$\displaystyle A(-4,7)$ and $\displaystyle B(8,-2)$ .
The point $\displaystyle C$ divides $\displaystyle AB$ in the ratio $\displaystyle {AC:CB}={2:1}$ .
Find the coordinates of $\displaystyle C$ .

\displaystyle x_C

/ – 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

\displaystyle y_C

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Work out the distance of the point $\displaystyle (-1,3)$ from the line $\displaystyle {5x-12y+2}={0}$ .

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Work out the distance between the lines $\displaystyle {y}={2x+1}$ and $\displaystyle {y}={2x-4}$ .
Write the result in its simplest form.

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Find the locus of all the points whose distance from the origin equals their distance from the point $\displaystyle (6,-2)$ .

\displaystyle O=(0,0)

\displaystyle (6,-2)

The triangle’s vertices: $\displaystyle A(0,0)$ , $\displaystyle B(6,0)$ , $\displaystyle C(3,9)$ .
$\displaystyle G$ is where the medians meet.

What are the coordinates of $\displaystyle G$ ?

Work out the distance of

\displaystyle G

from the line

\displaystyle AC

, in its simplest form.

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The distance of the point $\displaystyle (2,k)$ from the line $\displaystyle {3x+4y}={0}$ is $\displaystyle 4$ .
Find $\displaystyle k$ .

The base is

\displaystyle AB

, and the height – the distance of

\displaystyle C

from the line

\displaystyle AB

The triangle’s vertices: $\displaystyle A(1,1)$ , $\displaystyle B(4,5)$ , $\displaystyle C(6,-1)$ .
Work out the triangle’s area.

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The lines $\displaystyle {y}={ax+2}$ and $\displaystyle {6x-2y}={5}$ are parallel.

What is $\displaystyle a$ ?

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Put in the

\displaystyle a

you found, and solve the system.

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
$\displaystyle y=\pm\frac{b}{a}x$