Mathematics Coordinate Geometry Parabolas | Standard Equations | The Conic Sections and...

Parabolas | Standard Equations | The Conic Sections and Quadratic Equations

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Study of Conics gives us ability to analyze the motion that how to track the position of a moving object as a function of time.  Because, conics are the traveling path for the electrons, planets, satellites and other moving bodies whose motions are driven by inverse square forces. Once we know that the path of a moving body is a conic section, we immediately have information about the body’s velocity and the force that drives it. Circles, Parabolas, Ellipse, and Hyperbolas are the topics which we discuss in Conic Section. 

In today’s geometry conics are required to describe the graphs of Quadratic Equations in coordinate plane. The Greeks of Plato’s time described these curves as the curves formed by cutting a double cone with a plane. Hence the name conics come from.

Circles | Definition

A circle is the set of points in a plane whose distance from a given fixed point in the plane is constant.  The fixed point is the center of the circle, and the constant distance is the radius. The standard form of equations for circles derives from the distance formula: 
$$d= \sqrt{(x_{2}-x_{1})^2 + (y_{2}-y_{1})^{2}}$$
Equation of the circle of radius $a$, center at the origin: $x^{2}+y^{2}=a^{2}.$ 
Equation of circle of radius $a$, center at the point (h, k): $(x-h)^{2}+(y-k)^{2}=a^{2}.$

Parabolas | Definition | Standard Eqn

Parabola is a set of points in a plane equidistant from a given fixed point and a given fixed line in the plane. The fixed point is the Focus of the parabola and the fixed line is the Directrix.

Circle parabola ellipse hyperbola - CS School

If the focus F lies on directrix, the parabola is the line through F perpendicular to directrix. We consider this to be a degenerate case, and assume henceforth that F does not lies on directrix. A parabola has its simplest equation when

its Focus and Directrix straddle one of the coordinate axes. As in above figure: the Focus lies at the point F(0, p) on the positive y-axis. The directrix is the line y = -p. A point P(x, y) lies on the parabola if and only if PF = QF. From the distance formula,

$$PF = \sqrt{(x-0)^{2}+(y-p)^{2}} = \sqrt{x^{2}+(y-p)^{2}}$$

$$QF = \sqrt{(x-x)^{2}+(y-(-p))^{2}} = \sqrt{(y+p)^{2}}$$

Now, PF = QF Or, $\sqrt{x^{2}+(y-p)^{2}} = \sqrt{(y+p)^{2}}$ 
$$\text{Or, } x^{2} = 4yp.$$
This the Standard form of parabola. The point where a parabola crosses its axis is the vertex. The vertex of the parabola $x^{2}=4py$ lies at the origin. The positive number p is parabolas focal length.

If the parabola opens downward, with focus at (0, – p), and its directrix the line y = p, then equation of parabola become:

$$x^{2} = -4py$$

We obtain similar equations for parabolas opening to the right or to the left. Table below shows – standard form equations for parabolas with vertices at the origin:

Equations
Focus
Directrix
Axis
Opens
$x^{2}=4py$
$(0,p)$
$y= -p$
$y-axis$
UP
$x^{2}=-4py$
$(0, -p)$
$y= p$
$y-axis$
DOWN
$y^{2}=4py$
$(p,0)$
$x= -p$
$x-axis$
To the right
$y^{2}=-4py$
$(-p,0)$
$x= p$
$x-axis$
To the left
parabolas - CS School

Example: Find the Focus and Directrix of the parabola $y^{2}=10x.$

Solution: Comparing with the standard-form equation of parabola $y^{2}=4px:$
$4p = 10,$ therefore,
$$p=\frac{10}{4}, or, p=\frac{5}{2}.$$

So, the Focus F, of the parabola, $(p, 0) = (\frac{5}{2}, 0),$ and 
Directrix L: $x=-p, or, x=-\frac{5}{2}$

//This is the end of discussion on parabolas. Next, in this section >> we will discuss about ellipses and other conics, and their applications. 

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