Practice: the circle

Analytic geometry, grade 12 (5 units) Lesson 11 of 21Practice

Practice: the circle

The circle: $\displaystyle {x^{2}+y^{2}-10x+4y+20}={0}$ .
Complete the square, and find the centre and the radius.

/ – 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 and the radius:

The radius

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The point $\displaystyle (5,7)$ is on the circle $\displaystyle {x^{2}+y^{2}-4x-6y-12}={0}$ .
Write the equation of the tangent to the circle at this point.

\displaystyle x^{2}+y^{2}-4x-6y-12=0

\displaystyle (5,7)

Find where the line $\displaystyle {y}={x-1}$ and the circle $\displaystyle {(x-2)^{2}+(y-3)^{2}}={8}$ meet.

The circles $\displaystyle {x^{2}+y^{2}-2x-8}={0}$ and $\displaystyle {(x-1)^{2}+(y-5)^{2}}={4}$ .

How do the circles lie? Explain.

Find the common points.

Half the chord, the centre’s distance from the line and the radius make a right-angled triangle.

The centre of the circle is $\displaystyle (-1,4)$ .
The line $\displaystyle {4x-3y-9}={0}$ cuts the circle in a chord of length $\displaystyle 8$ .
Write the equation of the circle.

\displaystyle 4x-3y-9=0

\displaystyle (-1,4)

$\displaystyle A(0,0)$ and $\displaystyle B(8,0)$ .
Find the locus of the points three times as far from $\displaystyle B$ as from $\displaystyle A$ .

\displaystyle A=(0,0)

\displaystyle B=(8,0)

The circle $\displaystyle {x^{2}+y^{2}}={R^{2}}$ touches the line $\displaystyle {3x+4y}={20}$ .
Find $\displaystyle R$ .

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For which values of $\displaystyle n$ does the line $\displaystyle {y}={-x+n}$ cut the circle $\displaystyle {x^{2}+y^{2}}={18}$ at two points?

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