Form (1)
- Equation : $x^2=4py$
- Axis of Symmetry : $y-\text{axis}$
- Vertex at $(0,0)$
- Focus at $(0,p)$
- Directrix : $y = -p$
- If $p>0$, the parabola opens up
- If $p<0 $, the parabola opens down
Form (2)
- Equation : $y^2=4px$
- Axis of Symmetry : $x-\text{axis}$
- Vertex at $(0,0)$
- Focus at $(p,0)$
- Directrix : $x = -p$
- If $p>0$, the parabola opens to the right.
- If $p<0 $, the parabola opens to the left.
Latus Rectum
The focal chord which is perpendicular to the axis is known as latus rectum of the conic. Axis of symmetry ပေါ် ထောင့်မတ်ကျသော Focal Chord ကို latus rectum ဟုခေါ်သည်။For $y^2= 4px,\ \ p\ne 0 $,
When $x = p, y^2=4p^2$.
Therefore $y=\pm 2p$.
Focus = $(p, 0)$
The endpoints of latus rectum is $(p,-2p)$ and $(p,2p)$ .
And the length of latus rectum is $|4p|$.
For $x^2= 4py,\ \ p\ne 0 $,
Focus = $(0, p)$
When $y = p, x^2=4p^2$.
Therefore $x=\pm 2p$.
Hence the endpoints of latus rectum is $(-2p,p)$ and $(2p,p)$ .
And the length of latus rectum is $|4p|$.
Vertex က $(0,0)$ ၌ ရှိသော Parabola ပုံစံလေးခု၏ အနှစ်ချုပ်ကို အောက်ပါဇယားဖြင့်မှတ်သားနိုင်ပါသည်။
| Equation | Focus | Directrix | Axis of Symmetry | Endpoinds of Latus Rectum | Opens | |
|---|---|---|---|---|---|---|
$x^2=4py$ | $(0,p$) | $y=-p$ | $y-\text{axis}$ | ($\pm 2p, p)$ | UP when $p>0$ | |
| DOWN when $p<0$ | ||||||
$y^2=4px$ | $(p,0)$ | $x=-p$ | $x-\text{axis}$ | $(p,\pm 2p)$ | RIGHT when $p>0$ | |
| LEFT when $p<0$ | ||||||
Graphing Parabola
- to find the focus
- to find the equation of directrix
- to find the axis of symmetry
- to find the vertex
- to find the end points of latus rectum
- to draw the smooth curve passing through the vertex and the end points of latus rectum to form parabola
| Example (1) Graph $y^2 = 24x$. Solution Comparing with $y^2=4px$, we get $p=6$ Axis of symmetry : $x-\text{axis}$ The parabola opens to the right. Focus = $(p,0)=(6,0)$ $\therefore \ \ \ $ directrix : $x= -6$ Vertex = $(0,0)$ The endpoints of latus rectum = $(p,\pm 2p) = (6,\pm 12)$ Example (2) Graph $x^2 = -6y$. Solution Comparing with $x^2=4py$, we get $4p=-6\Rightarrow p=-\displaystyle\frac{3}{2}$ Axis of symmetry : $y-\text{axis}$ The parabola opens down. Focus = $ (0,p)=\left( {0,-\displaystyle\frac{3}{2}} \right)$ $\therefore \ \ \ $ directrix : $y= \displaystyle\frac{3}{2}$ Vertex = $(0,0)$ The endpoints of latus rectum = $ \left( {\pm 2p,\ p} \right)=\left( {\pm 3,\ -\displaystyle\frac{3}{2}} \right)$ |
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စာဖတ်သူ၏ အမြင်ကို လေးစားစွာစောင့်မျှော်လျက်!