Introduction
αိα်αျα်းαိုα်αားαော ααော့αျွα် (Cone) αှα ်αုαို αြα်αီαα ်αုαှα့် αိုα်းαြα်αိုα်αောα‘αါ ααှိαာαα့် α‘α ိα်α‘αိုα်းαျားαို Conic Sections αုαေါ်αα်။ αိုα်းαြα်αိုα်αော αြα်αီ၏ αα်αေαုံ (αောα့်α‘αေα‘αား) αို αိုα်၍ Conic Sections αျားαို
- Circle
- Ellipse
- Parabola
- Hyperbola
- Intersecting Lines
Parabola
- αေးαားαော applet αှိ α‘αှα် $P$ αို αွα်αွဲαြီး αွှေ့αြα့်αါ။
- α‘αှα် $P$ αα် α‘αှα်αေ (ααွေ့αျားαော α‘αှα်) $F$ αှα့် αျα်းαေ (ααွေ့αျားαော αျα်း) $L$ αှ α‘αြဲαူαီα ွာ αွားαေးαေαα်αို αွေ့ααါαα်။
- α‘αှα်αှα် (αွေ့αျားαိုα်αော α‘αှα်) $P$ αို αေαာαိုα်αြီး $P_1, P_2, P_3, \text{etc}.$ α ααြα့် α‘αα်αα်αှα်αα် αိုαါα ို့။ αα်αေαာααူαော $P$ α‘αှα်αျား၏ α‘α ုα‘αေးαို αျα်းαွေး (Curve) α‘αြα ် αွေ့αြα်ααα် αြα ်αြီး၊ α‘αိုαါ curve αို Parabola αုαေါ်αα်။
- α‘αှα်αေ (fixed point) $F$ αို Focus αုαေါ်αြီး၊ αျα်းαေ (fixed line) $L$ αို Directrix αုαေါ်αα်။
- αို့αြောα့် Parabola αိုαα်αှာ Focus αှα့် Directrix αို့αှ αူαီα ွာαေးαွာαα့် α‘αှα်αျားαါαα်αော α‘α ုαု α‘αိα္αါα်αα်αှα်αိုα်αါαα်။
| Definitions - Parabola A set that consists of all the points in a plane equidistant from a given fixed point and a given fixed line in the plane is a parabola. The fixed point is the focus of the parabola. The fixed line is the directrix. |
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Focus αα် $\text{positive}\ x-\text{axis}$ αေါ်αှိ α‘αှα်αα ်αု $(0,p)$ αြα ်αα်αု αူααα်။ Directrix αα် $(0,-p)$ αို αြα်αွားαော horizontal line (αေαြα်αီαျα်း) αြα ်αα်αု αူααα်။
αိုα‘αါ parabola αα် $(0,p)$ αှα့် $(0,-p)$ αို့၏ α‘αα်αှα်αြα ်αော $(0,0)$ αို αြα်αွားαα်။ ၎α်းαို parabola ၏ vertex αုαα်း αေါ်αα်။ Vertex αှα့် Focus αα် αူαီαော αောα်αိုα် αို့ααုα် αေαြα်αီ αျα်းαα ်αြောα်းαα်း αေါ်αွα်αာ αှိαα်။ Vertex αှα့် Focus αြားα‘αွာα‘αေး |p| αို parabola ၏ focal lenght αုαေါ်αα်။ Parabola ၏ vertex αို αα့်αα့်αြα်αြီး Parabola αို αေါα်αျိုးαီαα်αα်αိုα်း αွားαော $y-\text{axis}$ αို Axis of Symmetry (αေါα်αျိုးαီ αα်αိုး) αု αေါ်αα်။
α‘αှα် $P$ αှ directrix αေါ်αို့ αောα့်αα်αျαော αျα်းαα် directrix αို $Q$ ၌ αြα်αα်αိုαါα ို့။ Parabola ၏ α‘αိα္αာα်αα်αှα်αျα်α‘α $PF=PQ$ α‘αြဲαြα ်αါαα်။ αို့αြောα့် PF:PQ = 1 αြα ်αျှα် parabola αုαေါ်αα်။ α‘αိုαါ PF αှα့် PQ α‘αျိုးαို conic section αα ်αု၏ eccentricity ($e$) αု αα်αှα်αα်။
Distance Formula α‘α ...
$\begin{array}{l} P F=\sqrt{(x-0)^{2}+(y-p)^{2}}=\sqrt{x^{2}+(y-p)^{2}} \\\\ P Q=\sqrt{(x-x)^{2}+(y+p)^{2}}=\sqrt{(y+p)^{2}} \\\\ \therefore \sqrt{x^{2}+(y-p)^{2}}=\sqrt{(y+p)^{2}} \\\\ x^{2}+(y-p)^{2}=(y+p)^{2} \\\\ x^{2}=(y+p)^{2}-(y-p)^{2} \\\\ x^{2}=(y+p+y-p)(y+p-y+p) \\\\ x^{2}=(2 y)(2 p) \\\\ x^{2}=4 p y \text { or } y=\displaystyle\frac{x^{2}}{4 p} \end{array}$
αို့αြောα့် Focus α $(0,p)$ ၌ αှိαြီး directrix : $y=-p$ αြα ်αျှα် parabola ၏ equation αို $x^{2}=4 p y \text { or } y=\displaystyle\frac{x^{2}}{4 p}$ αု α‘αွα်ααူ αြောαိုα်αα်။ Parabola Shape αှာ Opens Up αြα ်αα်။
α‘αα်၍ Focus α $(0,-p)$ ၌ αှိαြီး directrix : $y=p$ αြα ်αျှα် parabola ၏ equation αှာ $x^{2}=-4 p y \text { or } y=-\displaystyle\frac{x^{2}}{4 p}$ αြα ်αြီး Opens Down Parabola αို ααှိαα် αြα ်αα်။ α‘ောα်αါ applet αှာ α‘αှα် $P$ αို αွα်αွဲαြα့်αါ။
α‘αα်၍ Focus α $(p,0)$ ၌ αှိαြီး directrix : $x=-p$ αြα ်αျှα် parabola ၏ equation αှာ $y^{2}=4px \text { or } x=\displaystyle\frac{y^{2}}{4 p}$ αြα ်αြီး Opens Right Parabola αို ααှိαα် αြα ်αα်။α‘ောα်αါ applet αှာ α‘αှα် $P$ αို αွα်αွဲαြα့်αါ။
αိုαα်းαူα ွာ Focus α $(-p,0)$ ၌ αှိαြီး directrix : $x=p$ αြα ်αျှα် parabola ၏ equation αှာ $y^{2}=-4px \text { or } x=-\displaystyle\frac{y^{2}}{4 p}$ αြα ်αြီး Opens Left Parabola αို ααှိαα် αြα ်αα်။ αော်αြαါ parabola αုံαα္αာα် αေးαျိုးαို α‘ောα်αါα‘αိုα်းαှα်αားαိုα်αါαα်။
Standard Equations
Standard-form equations for parabolas with vertices at the origin (p > 0)
| Equation | Focus | Directrix | Axis | Opens |
|---|---|---|---|---|
| $x^{2}=4 p y$ | $(0, p)$ | $y=-p$ | $y \text { -axis }$ | UP |
| $x^{2}=-4 p y$ | $(0, -p)$ | $y=p$ | $y \text { -axis }$ | DOWN |
| $y^{2}=4 p x$ | $(p, 0)$ | $x=-p$ | $x \text { -axis }$ | RIGHT |
| $y^{2}=-4 p x$ | $(-p, 0)$ | $x=p$ | $x \text { -axis }$ | LEFT |
Terms of Parabola
Important Terms to Know
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Worked Examples
| Example (1) Find the equation of the parabola with focus at (0, 4) and directrix the line y=-4. Solution Focus = (0,4), directrix : y= -4 Since the focus is on the positive y-axis at (0, 4) and the directrix is a horizontal line y=-4, the vertex of the parabola is (0,0). The equation of the parabola is of the form $x^2=4py$ where $p=4$. $\therefore \ \ x^2=4(4)y\ \ \ \text{or}\ \ \ y=\displaystyle\frac{x^2}{16}$ Example (2) Find the focus and directrix of the parabola $y^2= 10x$. Solution Parabola: $y^2= 10x$. Comparing with $y^2= 4px$, $\begin{array}{l} 4p=10\\\\ p=\displaystyle\frac{10}{4}=\displaystyle\frac{5}{2}\\\\ \therefore \ \ \ \text{focus}= (p,0)= \displaystyle \left( {\frac{5}{2},0} \right)\\\\ \ \ \ \ \ \ \ \ \text{directrix}: x=-p \Rightarrow x= -\displaystyle\frac{5}{2} \end{array}$ Example (3) Find the equation of a parabola whose vertex is at (0, 0). If the axis of symmetry of the parabola is the $x$-axis and the point $\displaystyle \left( {-\frac{1}{2},2} \right)$ lies on the graph. Find its focus and directrix, and graph the equation. Solution vertex = (0,0) axis of symmetry = $x-\text{axis}$ point on the graph = $ \left( {-\displaystyle\frac{1}{2},2} \right)\Rightarrow 2^{\text{nd}}\ \text{quadrant}$ $\therefore$ The parabola opens to the left and its equation is of the form $y^2=-4px$ Since $ \left( {-\displaystyle\frac{1}{2},2} \right)$ lies on parabola, ${2}^2=-4p \left( {-\displaystyle\frac{1}{2},2} \right)$ $\therefore\ \ p=2$ $\therefore$ The equation of parabola : $y^2=-4(2)x\Rightarrow y^2=-8x$ $ \ \ \ \ \ \ \text{focus} = (-p,0) = (-2,0)$ $ \ \ \ \ \ \ \text{directrix} : x = p\Rightarrow x =2 $ $ \displaystyle \begin{array}{l}\text{When}\ x=-2,\ {{y}^{2}}=-8(-2)=16\\\\ \therefore \ \ y=\pm 4\end{array}$ Hence the parabola passes through the points $(-2,4)$, $\displaystyle \left( {-\frac{1}{2},2} \right)$, $(0,0)$ and $(-2,-4)$.  |
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Exercises
| Find the focus and directrix of each of the foloowing parabola equations and sketch the graph. $\begin{array}{ll} (1) & y^{2}=12 x\\\\ (2) & x^{2}=6 y\\\\ (3) & x^{2}=-8 y\\\\ (4) & y^{2}=-2 x\\\\ (5) & y=4 x^{2}\\\\ (6) & y=-8 x^{2}\\\\ (7) & x=-3 y^{2}\\\\ (8) & x=2 y^{2} \end{array}$ |
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