A hyperbola with a line and a circle

Analytic geometry, grade 12 (5 units) Lesson 18 of 21Learning

A hyperbola with a line and a circle

A line and a hyperbola

Here too we solve the system. A line and a hyperbola may have two common points, one or none, and there are two special cases:

A line parallel to an asymptote meets the hyperbola at only one point and crosses it – it is no tangent. After substituting, $\displaystyle x^{2}$ cancels and a first-degree equation is left.
A line through the centre whose slope is steeper than the asymptotes’ does not meet the hyperbola at all.

An example: $\displaystyle {x^{2}-y^{2}}={16}$ and $\displaystyle {y}={x-2}$ – parallel to the asymptote $\displaystyle {y}={x}$ .

x^{2}-(x-2)^{2}=16\quad\Rightarrow\quad4x-4=16\quad\Rightarrow\quad x=5

One point: $\displaystyle (5,3)$ .

\displaystyle x^{2}-y^{2}=16

\displaystyle y=x-2

\displaystyle y=x

\displaystyle y=-x

\displaystyle (5,3)

A line touching a hyperbola

When substituting gives a quadratic equation, the line touches when its discriminant is $\displaystyle 0$ .

An example: for which values of $\displaystyle n$ does the line $\displaystyle {y}={2x+n}$ touch the hyperbola $\displaystyle {\frac{x^{2}}{4}-y^{2}}={1}$ ?

$\displaystyle {x^{2}-4(2x+n)^{2}}={4}$ , that is $\displaystyle {-15x^{2}-16nx-4n^{2}-4}={0}$ .
The discriminant: $\displaystyle {256n^{2}-4\cdot15\cdot(4n^{2}+4)}={16n^{2}-240}$ .
$\displaystyle {16n^{2}}={240}$ , so $\displaystyle {n}={\pm\sqrt{15}}$ .

A circle and a hyperbola

A circle with its centre on the $\displaystyle x$ -axis and a canonical hyperbola: as with the ellipse, isolate $\displaystyle y^{2}$ in one equation and put it into the other.

An example: $\displaystyle {x^{2}-y^{2}}={7}$ and $\displaystyle {x^{2}+y^{2}}={25}$ . Add the equations: $\displaystyle {2x^{2}}={32}$ , so $\displaystyle {x}={\pm4}$ , and $\displaystyle {y^{2}}={9}$ . Four points: $\displaystyle (\pm4,\pm3)$ .

\displaystyle x^{2}-y^{2}=7

\displaystyle x^{2}+y^{2}=25

Exercise 1

Find the points common to the hyperbola $\displaystyle {x^{2}-y^{2}}={9}$ and the line $\displaystyle {y}={2x-6}$ , and mark them on the board.

\displaystyle x^{2}-y^{2}=9

\displaystyle y=2x-6

Exercise 2

The hyperbola $\displaystyle {x^{2}-y^{2}}={9}$ and the line $\displaystyle {y}={2x}$ .

Explain the result with the slopes of the asymptotes.

Exercise 3

For which values of $\displaystyle n$ does the line $\displaystyle {y}={x+n}$ touch the hyperbola $\displaystyle {\frac{x^{2}}{4}-\frac{y^{2}}{3}}={1}$ ?

Exercise 4

Find the points common to the hyperbola $\displaystyle {\frac{x^{2}}{16}-\frac{y^{2}}{9}}={1}$ and the circle $\displaystyle {x^{2}+y^{2}}={25}$ .

The circle passes through the hyperbola’s foci. How can you see this from the equations?

Exercise 5

The line $\displaystyle {y}={\frac{3}{4}x+1}$ and the hyperbola $\displaystyle {\frac{x^{2}}{16}-\frac{y^{2}}{9}}={1}$ . Without solving: at most how many common points can they have? Why?

/ – 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

Formulas: analytic geometry, grade 12 (5 units)

Open as a page

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