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General Locus Problems


Defining a locus and finding an equation 

A locus is a pathway (perhaps a line or a curve) formed by a collection of points, each of which has a location that satisfies some given condition. 

For example, a straight line can be defined as the pathway that consists of all points satisfying an equation of the form $y=mx+b$y=mx+b. Points that are not on this line will not satisfy this equation and so are not part of the locus. 

In the diagram the red points satisfy the line $y=\frac{1}{2}x+3$y=12x+3 as exemplified by the point $\left(12,9\right)$(12,9), whereas the blue points, such as the point $\left(2,2\right)$(2,2), don't. 

In the study of cartesian geometry, we are often tasked with constructing an equation of a line or a curve that is based on some imposed condition. 

For example, suppose we wish to find the locus of all points $\left(x,y\right)$(x,y) that are at a distance of exactly $5$5 units from the origin. Just like a child on a play horse on the outer rim of a merry-go-round the locus becomes a circle with centre $\left(0,0\right)$(0,0) and radius $5$5.

From the distance formula, we can easily locate a point, say  $\left(3,4\right)$(3,4), that satisfies the imposed condition:

$d$d $=$= $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$(x2x1)2+(y2y1)2
  $=$= $\sqrt{\left(3-0\right)^2+\left(4-0\right)^2}$(30)2+(40)2
  $=$= $\sqrt{9+16}$9+16
  $=$= $\sqrt{25}$25
$\therefore$     $d$d $=$= $5$5

But what about any point $\left(x,y\right)$(x,y) on the locus? 

Again, using the distance formula, and setting $d=5$d=5, we have:

$5$5 $=$= $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$(x2x1)2+(y2y1)2
$5$5 $=$= $\sqrt{\left(x-0\right)^2+\left(y-0\right)^2}$(x0)2+(y0)2
$5$5 $=$= $\sqrt{x^2+y^2}$x2+y2
$\therefore$     $x^2+y^2$x2+y2 $=$= $25$25

This is such an interesting result. For example, the left hand side of this equation is symmetrical - we could swap values for $x$x and $y$y around and it would not make any difference to the sum. Hence we know $\left(3,4\right)$(3,4) and $\left(4,3\right)$(4,3) will both be on the circle.

Because the left hand side is the sum of two squares, we also know that $\left(-3,4\right)$(3,4) $\left(-3,-4\right)$(3,4)$\left(-4,3\right)$(4,3) and $\left(-4,-3\right)$(4,3) satisfy the equation. The shape of the locus reflects this symmetric relationship between $x$x and $y$y

Of course there are an infinite set of points $\left(x,y\right)$(x,y) that satisfy $x^2+y^2=25$x2+y2=25. Points like $\left(0,5\right)$(0,5)$\left(0,-5\right)$(0,5) and $\left(\sqrt{10},\sqrt{15}\right)$(10,15) for example. 

In general terms, the locus of all points lying a distance $r$r from the origin is given by $x^2+y^2=r^2$x2+y2=r2. Even more generally, the locus of all points lying a distance $r$r units from the fixed point $\left(h,k\right)$(h,k) is given, through the same process, by $\left(x-h\right)^2+\left(y-k\right)^2=r^2$(xh)2+(yk)2=r2.


Other examples

Example 1  

Find the equation of the locus of all points $7$7 units from the $x$x axis.

Here the locus is simply the two lines that are parallel to the $x$x axis and a distance of $7$7 away from it. The locus includes all points on both of the lines given by $y=\pm7$y=±7.

Example 2

Sometimes we talk about a variable point $\left(x,y\right)$(x,y) that moves according to the imposed condition, just like a train moves according to where the train tracks are pointing. Here is an example:

Find the locus of a variable point which moves so that its distance from the $y$y axis is always $3$3 times its distance from the $x$x axis. 

The diagram makes it clear that the value of the $x$x component must remain $3$3 times the value of the $y$y component. That is to say, for the point $\left(x,y\right)$(x,y), we have $x=3y$x=3y, and so the locus becomes the line given by  $y=\frac{1}{3}x$y=13x.

Example 3

Find the locus of the point which moves such that it remains equidistant from the two fixed points $\left(5,3\right)$(5,3) and $\left(2,8\right)$(2,8).

Here, we have to keep the variable point $\left(x,y\right)$(x,y) the same distance away from the given fixed points.

Using the distance formula, this means setting:

 $\sqrt{\left(x-5\right)^2+\left(y-3\right)^2}=\sqrt{\left(x-2\right)^2+\left(y-8\right)^2}$(x5)2+(y3)2=(x2)2+(y8)2 and simplifying:

$\sqrt{\left(x-5\right)^2+\left(y-3\right)^2}$(x5)2+(y3)2 $=$= $\sqrt{\left(x-2\right)^2+\left(y-8\right)^2}$(x2)2+(y8)2
$\left(x-5\right)^2+\left(y-3\right)^2$(x5)2+(y3)2 $=$= $\left(x-2\right)^2+\left(y-8\right)^2$(x2)2+(y8)2
$x^2-10x+25+y^2-6y+9$x210x+25+y26y+9 $=$= $x^2-4x+4+y^2-16y+64$x24x+4+y216y+64
$-10x-6y+34$10x6y+34 $=$= $-4x-16y+68$4x16y+68
$10y$10y $=$= $6x+34$6x+34
$y$y $=$= $\frac{3}{5}x+3\frac{2}{5}$35x+325

Thus the locus is a line with gradient $m=\frac{3}{5}$m=35 and $y$y-intercept $3\frac{2}{5}$325 with the general form given as $3x-5y+17=0$3x5y+17=0.

More worked examples

Question 1

Question 2

Question 3




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