Question

find the locus of the point, the absolute value of the difference of the distances of which from the points (2,2) and (0,0) is 2. identify the curve represented by the locus

Answer 1

Let (x,y) be the point whose locus is required. It is given that the distance from the points (0,0) and (2,2) is the same.

Let $d_1$ be the distance of (x,y) from (0,0). Then by distance formula, we will have:

$d_1=\sqrt{(x-0)^2+(y-0)^2}=\sqrt{x^2+y^2}$....Equation 1

Now, if the above distance is 2, we have: $\sqrt{x^2+y^2}=2$

or, $x^2+y^2=4$

$\therefore y=4-x^2$.......Equation 2

Likewise, let $d_2$ be the distance of (x,y) from (2,2). Then, again, by distance formula we will have:

$d_2=\sqrt{(x-2)^2+(y-2)^2}$..............Equation 3

Since, both the distances are equal to 2, we have:

$d_1=d_2$

$\sqrt{x^2+y^2}=\sqrt{(x-2)^2+(y-2)^2}$

${x^2+y^2}={(x-2)^2+(y-2)^2}$

$x^2+y^2=x^2+4-4x+y^2+4-4y$

$4x+4y=8$

$y=2-x$...........Equation 4

Now, plugging in $y=\sqrt{4-x^2}$ from Equation 2 in Equation 4 we get:

$\sqrt{4-x^2} =2-x$

Squaring both sides we will get:

$4-x^2=(2-x)^2=4+x^2-4x$

Or, $4x-2x^2=2x(2-x)=0$

Therefore, either x=0 or x=2.

Now, when x=0, y=2 and when x=2, y=0.

Therefore, the curve represented by the locus is a straight line with the intercept 2 and slope 1 as: $y=x+2$

Please find the graph in the attached file for a detailed understanding of the solution.

Answer 2

A locus is the set of all points generally forming a curve or surface satisfying several conditions.

For example, the locus of points in the plane equidistant from a specified point is a circle, and the set of points in three-space equidistant from a specified point is a sphere.