aisha said:
OK yes I do understand what a locus is now but how do I come up with an equation for the locus?
There is no set way of doing this, each problem is different, Generally you have to let your mathematical intuition guide you to the solution. I'll go through this problem in detail as an example.
In this problem,
You are given the condition that every point in the locus must be 3 units from (-1,2). Intuition should tell you that the locus should form a circle since a circle is defined to be the set of points in a plane that are equidistant from a fixed point. That distance is called the radius. However, knowing this ahead of time could save a little effort, but it is not mandatory to solving the problem.
So let's forget about circles for now, our goal is to come up with an equation for every point of the locus. Let's start with what's given and see what we can come up with. I'll restate the given just to emphasize its importance.
1. were given the point (-1,2)
2. were told that every point in the locus must be 3 units from that point.
Now if one doesn't know the distance formula, then this problem is virtually unsolvable. You could discover it on your own from the pythagorean theorem, but if a student at that level doesn't know the distance formula, then chances are, they wouldn't know the pythagorean theorem either. Bottom line, you have to start somehwere from the bag of tools you've accumulated thus far, and figure out what to use by what your given information is. Since were given distance between points, we would think to try the distance formula.
Another way of saying a point is a distance of 3 units from (-1,2), is to say that it should satisfy the distance equation; That is we should get
\sqrt{ [x-(-1)]^2 + [y-2]^2 } = 3
when we plug in the coordinates of the point (x,y) for a point in the locus.
since we want an equation for
all such points, well leave x and y arbitrary (that is we won't specify a specific value for x and y).
Now technically at this point were done. We've come up with an equation that describes all the points for the locus. It explicitly states that if a point satisfies this equation, then it is in the locus, if it doesn't satisfy the equation then it is not in the locus since it would not be a distance of 3 from (-1,2).
To make it look nicer, and for reason's you'll see further down this post, we can square both sides to obtain the result
(x+1)^2 + (y-2)^2 =9
your result is correct!
the equation of a circle is r^2=x^2+y^2
I found an example in my textbook the above equation is when the center of the circle is at (0,0) which is not the case in this question.
I have come up with an equation I don't know if this is correct it is
(x+1)^2 + (y-2)^2 =9 If this is correct how do I check myself?
Your result is correct. To check yourself, recall that we think this should be the equation of a circle.
The more general equation of a circle is defined to be.
(x-h)^2 + (y-k)^2 = r^2
with center at point (h,k) and radius r.
notice that when we plug in the values of h=0 and k=0 in this equation we get the same equation you found for a circle with center at (0,0). The more general equation of a circle can actually be derived from the distance equation. (just use h and k for the second coordinate ).
looking at your result, we see that it can be rewritten to resemble the equation of a circle
[x-(-1)]^2 + [y-2]^2 = 3^2
comparing this to the equation of a circle, we see that the result is an equation for a cirlce with center (-1,2) and radius 3. So our intuition was right.
The original question was describe the locus of points that are 3 units from (-1,2) What will my therefore statement be how do I answer the question?
To write the answer, it is enough to say that the locus is the set of all points that satisfy the equation,
(x+1)^2 + (y-2)^2 =9
or alternatively you could say that the locus forms a cirlce with the above equation.
I apologize that this post was so long, but I was trying to give an in depth approach to the thought process behind solving such a problem. In so doing I repeated myself many times but it seemed this was necessary due to the nature of the questions being asked. It seemed apparent that short answers were not having much of an effect.
-MS
