5.
1 The Parabola
Definition of a Parabola
The parabola is defined as the locus of a point which moves so that it is always the same distance from a fixed point (called the focus) and a given line (called the directrix ). In the following graph,
The focus of the parabola is at ( , p). The directrix is the line y = -p. The focal distance is !p! (Distance from the origin to the focus, and from the origin to the directrix. "e ta#e absolute value because distance is positive.) The point (x, y) represents any point on the curve. The distance d from any point (x, y) to the focus ( , p) is the same as the distance from (x, y) to the directrix.
$The word locus means the set of points satisfying a given condition. %ee some bac#ground in Distance from a Point to a &ine.'
The (ormula for a Parabola ) *ertical +xis
+dding to our diagram from above, we see that the distance d , y + p.
-ow, using the distance formula on the general points ( , p) and (x, y), and e.uating it to our value d , y + p, we have
%.uaring both sides gives/
(x 0)2 + (y p)2 = (y + p)2
%implifying gives us the formula for a parabola/
x2 = 4py
In more familiar form, with 0y = 0 on the left, we can write this as/
where p is the focal distance of the parabola.
&I*12ath
The &ive2ath graph is similar to the following. 1ach of the colour)coded line segments is the same length in this spider)li#e graph/
%#etch the parabola
(ind the focal length and indicate the focus and the directrix on your graph.
+nswer
+rch 3ridges 4 +lmost Parabolic
The 5ladesville 3ridge in %ydney, +ustralia was the longest single span concrete arched bridge in the world when it was constructed in 6789. The shape of the arch is almost parabolic, as you can see in this image with a superimposed graph of y = x2 (The negative means the legs of the parabola face downwards.)
$+ctually, such bridges are normally in the shape of a catenary, but that is beyond the scope of this chapter.'
Parabolas with :ori;ontal +xis
"e can also have the situation where the axis of the parabola is hori;ontal/
In this case, we have the relation:
y2 = 4px
$In a relation, there are two or more values of y for each value of x. <n the other hand, a function only has one value of y for each value of x.'
1xample ) Parabola with :ori;ontal +xis %#etch the curve and find the e.uation of the parabola with focus (2,0) and directrix x = 2.
+nswer
%hifting the *ertex of a Parabola from the <rigin
This is a similar concept to the case when we shifted the centre of a circle from the origin. To shift the vertex of a parabola from ( , ) to ( h, k), each x in the e.uation becomes (x 4 h) and each y becomes (y 4 k). %o if the axis of a parabola is vertical , and the vertex is at (h, k), we have
(x h)2 = 4p(y k)
&et=s see what this means in &ive2ath/
&I*12ath
If the axis of a parabola is horizontal, and the vertex is at (h, k), the e.uation becomes
(y k)2 = 4p(x h)
1xercises 6. %#etch x2 , 69y
+nswer
>. (ind the e.uation of the parabola having vertex (0,0), axis along the x)axis and passing through (2,-1).
+nswer
?. "e found above that the e.uation of the parabola with vertex (h, k) and axis parallel to the y)axis is (x 4 h)2 , 9p(y 4 k). %#etch the parabola for which (h, k) is (-1,2) and p , )?.
+nswer
%ee also/ :ow to draw y2 = x - 2@
+pplications of Parabolas
+pplication 6 ) +ntennas
+ parabolic antenna has a cross)section of width 6> m and depth of > m. "here should the receiver be placed for best reception@
+nswer
+pplication > ) ProAectiles + golf ball is dropped and a regular strobe light illustrates its motion as follows...
"e observe that it is a parabola. ("ell, very close). "hat is the e.uation of the parabola that the golf ball is tracing out@
+nswer
MINIST ! "# $%&'(TI"N M()(!SI( *achelor of Teaching +ith ,onors *(',$)" "# T$(',IN- %$- $$ P "- (MM$ .P IM( ! $%&'(TI"N/
-$"M$T ! T I() $0(M '"%$ N" 1 MT$2132 N(M$ 144444444444444444444444444444444 %ate 1 2 N"5$M*$ 6313 %uration 1 67 hours
INST &'TI"NS T" '(N%I%(T$S 6. This examination paper consists of three sections. %ection + %ection 3 %ection B >. (i) (ii) ?. / / / 63 obAective .uestions 8 structured .uestions 2 essay .uestions
+nswer all the .uestions in %ection + and %ection 3. +nswer > out of ? .uestions in %ection B.
+ll answer sheet of %ection +, %ection 3 and %ection B must be tied together with this front page and handed in at the end of the examination.
FOR EXAMINER USE ONLY S$'TI"N ( 63 <ar=s * 83 <ar=s ' 83 <ar=s 9&$STI"N N" : M( ;S 1 > 63 1.a/ 1.ba 6.a/ 6.b/ 2.a/ 2.b/ 8.a/ 8.b/ T"T()
1.a/
1.b/
1.c/
1.d/
6.a/
6.b/
2.a/
2.b/
- (N% T"T()
-$"M$T ! MT$ 2132 (NS?$ S,$$T .S$'TI"N (/
N(M$ 1 @@@@@@@@@@@@@@@@@@@@@@@@@@@@ 1. 6. 2. 8. 5. A. B. C. D. 13. 11. 16. 12. 18. 15. 1A. 1B. 1C. 1D. 63. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( * * * * * * * * * * * * * * * * * * * * ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' % % % % % % % % % % % % % % % % % % % %
