Question

the orientarion of a satelite dish thar monitors radio waves modelled by the equation
${2x}^{2} + \sqrt{3} xy + {y}^{2} - 20 = 0$
​

Answer 1

Step-by-step explanation:

focus, is equal to the distance from a fixed line, called the directrix. The point halfway between the focus and the directrix is called the vertex of the parabola.

A parabola is drawn with vertex at the origin and opening up. A focus is drawn as F at (0, p). A point P is marked on the line at coordinates (x, y), and the distance from the focus to P is marked d. A line marked the directrix is drawn, and it is y = − p. The distance from P to the directrix at (x, −p) is marked d.

Figure 10.5.3: A typical parabola in which the distance from the focus to the vertex is represented by the variable p.

A graph of a typical parabola appears in Figure 10.5.3. Using this diagram in conjunction with the distance formula, we can derive an equation for a parabola. Recall the distance formula: Given point P with coordinates (x1,y1) and point Q with coordinates (x2,y2), the distance between them is given by the formula

d(P,Q)=

√

(x2−x1)2+(y2−y1)2

.

Then from the definition of a parabola and Figure 10.5.3, we get

d(F,P)=d(P,Q)

√

(0−x)2+(p−y)2

=

√

(x−x)2+(−p−y)2

.

Squaring both sides and simplifying yields

x2+(p−y)2=02+(−p−y)2 x2+p2−2py+y2=p2+2py+y2 x2−2py=2py x2=4py.

Now suppose we want to relocate the vertex. We use the variables (h,k) to denote the coordinates of the vertex. Then if the focus is directly above the vertex, it has coordinates (h,k+p) and the directrix has the equation y=k−p. Going through the same derivation yields the formula (x−h)2=4p(y−k). Solving this equation for y leads to the following theorem.

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