solve problems using loci
A.Find the number of points that are three units from the origin and also two units from the x-axis.
B.Give an equation or equations that describe the locus of pts:
1) equidistant from the lines x= -1 and x=5
2) equidistant from the liens y=-3 and y=-7
3) equidistant from the lines y= 3x+1 and y=3x+9
A) Points 3 units from the origin are on the circle x^2 + y^2 = 3^2 = 9.
The second condition implies that y = ± 2.
Plugging in either value of y yields x^2 + 4 = 9 ==> x = ±√5.
So, we have four points: (x,y) = (±√5, 2), (±√5, -2).
1) These lines are both vertical. So, the line equidistant from them is vertical, too.
The midpoint of -1 and 5 is its average (-1 + 5)/2 = 2.
So, the locus is x = 2.
2) These lines are both horizontal. So, the line equidistant from them is horizontal, too.
The midpoint of -3 and -7 is its average (-3 + -7)/2 = -5.
So, the locus is y = -5.
3) These lines both have slope 3 (and are parallel).
So, the locus is a line with the same slope half way between:
Use the y-intercepts' midpoint: (1 + 9)/2 = 5.
So, the locus is y = 3x + 5.
A locus is a line that is always the same distance from another line, or point.
A circle, for example, is a locus of its centre - and is commonly used to solve loci problems. (A compass is your best friend in solving loci problems!)
When something is the same distance from two or more other things (lines, points, objects), we say that it is equidistant from those things (i.e. Equal-Distant).
For example, in the following image, the red line is equidistant from the two black lines.
You will be expected to be able to construct a range of standard loci, as well build loci for unfamiliar shapes, based on the rules you know.