Practice: the parabola and the ellipse

Analytic geometry, grade 12 (5 units) Lesson 16 of 21Practice

Practice: the parabola and the ellipse

A parabola’s focus is $\displaystyle (-3,0)$ , and its directrix $\displaystyle {x}={3}$ .
Write the parabola’s equation.

\displaystyle x=3

\displaystyle (-3,0)

Write the tangent to the parabola $\displaystyle {y^{2}}={4x}$ at the point $\displaystyle (4,-4)$ .

\displaystyle y^{2}=4x

\displaystyle (4,-4)

The parabola $\displaystyle {y^{2}}={4x}$ and the line $\displaystyle {y}={x+1}$ .

How do the line and the parabola lie?

An ellipse’s vertices on the $\displaystyle y$ -axis are $\displaystyle (0,\pm5)$ , and its foci $\displaystyle (0,\pm4)$ .
Write the equation of the ellipse.

\displaystyle (0,4)

\displaystyle (0,-4)

The ellipse $\displaystyle {\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}}={1}$ passes through the point $\displaystyle (2,3)$ .

Find

\displaystyle b^{2}

: write an equation with

\displaystyle k

in place of

\displaystyle b^{2}

Write the equation of the ellipse.

\displaystyle (2,3)

Find the points common to the ellipse $\displaystyle {x^{2}+4y^{2}}={20}$ and the line $\displaystyle {x+2y}={6}$ .

Find the points common to the circle $\displaystyle {x^{2}+y^{2}}={9}$ and the parabola $\displaystyle {y^{2}}={8x}$ .

The point $\displaystyle P(3,y_P)$ , with $\displaystyle y_P>0$ , is on the ellipse $\displaystyle {\frac{x^{2}}{25}+\frac{y^{2}}{9}}={1}$ .
$\displaystyle F_1(-4,0)$ and $\displaystyle F_2(4,0)$ are the foci.

\displaystyle y_P

/ – 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 PF_2

/ – 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 PF_1

– it can be done without roots.

/ – 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

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