Question 315051
find all points having an x coordinate of 9, whose distance from the point (3,-2) is 10
<pre><b>
All points which have x coordinate 9 are on the vertical
green line below:

{{{drawing(400,400,-3,17,-12,8, locate(.5,-2,"(3,-2)"),
line(3-.1,-2,3+.1,-2), line(3,-2-.1,3,-2+.1),
graph(400,400,-3,17,-12,8), green(line(9,-11,9,11)) )}}}

we take a compass and open it up to a radius of 10 units.
We put the sharp point of the compass on the point (3,-2) 
and we find that can swing two red arc to cut the green vertical
line in two points, like this:

{{{drawing(400,400,-3,17,-12,8, locate(.5,-2,"(3,-2)"),
line(3-.1,-2,3+.1,-2), line(3,-2-.1,3,-2+.1),
graph(400,400,-3,17,-12,8), green(line(9,-11,9,11)),

red(arc(3,-2,20,-20,40,70)), red(arc(3,-2,20,-20,290,320)) 


)}}}

Then we draw 10-unit long blue lines from (3,-2) to the 
points where the arcs cut the green line:

{{{drawing(400,400,-3,17,-12,8, locate(.5,-2,"(3,-2)"),
line(3-.1,-2,3+.1,-2), line(3,-2-.1,3,-2+.1),
graph(400,400,-3,17,-12,8), green(line(9,-11,9,11)),
blue(line(3,-2,9,6)), blue(line(3,-2,9,-10)),
red(arc(3,-2,20,-20,40,70)), red(arc(3,-2,20,-20,290,320)),
locate(9,6.5,"(9,??)"), locate(9,-10,"(9,??)"), 
locate(5,2.5,10), locate(5,-5.7,10)

)}}}

Those two points look like they are (9,6)  and (9,-10), but
looking and estimating is not good enough for mathematics.
We must calculate them:

Let's draw a black horizontal line segment from (3,-2) over to the
green vertical line:

{{{drawing(400,400,-3,17,-12,8, locate(.5,-2,"(3,-2)"), line(3,-2,9,-2), 
line(3-.1,-2,3+.1,-2), line(3,-2-.1,3,-2+.1), locate(9,-2,"(9,-2)"),
graph(400,400,-3,17,-12,8), green(line(9,-11,9,11)),
blue(line(3,-2,9,6)), blue(line(3,-2,9,-10)),
red(arc(3,-2,20,-20,40,70)), red(arc(3,-2,20,-20,290,320)),
locate(9,6.5,"(9,??)"), locate(9,-10,"(9,??)"), locate(6,-2,6),
locate(5,2.5,10), locate(5,-5.7,10)
)}}}

That line segment is 6 units long because it extends horizontally
from the point (3,-2) over to (9,-2) and that is 6 units.  

{{{drawing(400,400,-3,17,-12,8, locate(.5,-2,"(3,-2)"), line(3,-2,9,-2), 
line(3-.1,-2,3+.1,-2), line(3,-2-.1,3,-2+.1), locate(9,-2,"(9,-2)"),
graph(400,400,-3,17,-12,8), green(line(9,-11,9,11)),
blue(line(3,-2,9,6)), blue(line(3,-2,9,-10)),
red(arc(3,-2,20,-20,40,70)), red(arc(3,-2,20,-20,290,320)),
locate(9,6.5,"(9,??)"), locate(9,-10,"(9,??)"), locate(6,-2,6),
locate(5,2.5,10), locate(5,-5.7,10)
)}}} 

If we look just at the top half, we have a right triangle, with
hypotenuse 10 units and horizontal leg 6 units, like this.  Call the
length of the green side of the right triangle h,

{{{drawing(400,400,-3,17,-12,8, locate(.5,-2,"(3,-2)"), line(3,-2,9,-2), 
line(3-.1,-2,3+.1,-2), line(3,-2-.1,3,-2+.1), locate(9,-2,"(9,-2)"),
graph(400,400,-3,17,-12,8), green(line(9,-2,9,11)),
blue(line(3,-2,9,6)), locate(9.3,2,h),
red(arc(3,-2,20,-20,40,70)), 
locate(9,6.5,"(9,??)"), locate(6,-2,6),
locate(5,2.5,10)
)}}}

By the Pythagorean theorem

{{{6^2 + h^2 = 10^2}}}
{{{36+h^2=100}}}
{{{h^2=100-36}}}
{{{h^2=64}}}
{{{h=sqrt(64)}}}
{{{h=8}}}

So the upper vertex of that right triangle
is 8 units above (9,-2) so it must be (9,6),
just as we guessed just by looking.

Putting back the lower right triangle, which is congruent to
the upper one:

{{{drawing(400,400,-3,17,-12,8, locate(.5,-2,"(3,-2)"), line(3,-2,9,-2), 
line(3-.1,-2,3+.1,-2), line(3,-2-.1,3,-2+.1), locate(9,-2,"(9,-2)"),
graph(400,400,-3,17,-12,8), green(line(9,-11,9,11)),
blue(line(3,-2,9,6)), blue(line(3,-2,9,-10)), locate(9.3,2,"h=8"),
red(arc(3,-2,20,-20,40,70)), red(arc(3,-2,20,-20,290,320)),
locate(9,6.5,"(9,6)"), locate(9,-10,"(9,??)"), locate(6,-2,6),
locate(5,2.5,10), locate(5,-5.7,10)
)}}}

So the lower vertex of that lower right triangle
is 8 units below (9,-2) so it must be (9,-10),
just as we guessed just by looking.

{{{drawing(400,400,-3,17,-12,8, locate(.5,-2,"(3,-2)"), line(3,-2,9,-2), 
line(3-.1,-2,3+.1,-2), line(3,-2-.1,3,-2+.1), locate(9,-2,"(9,-2)"),
graph(400,400,-3,17,-12,8), green(line(9,-11,9,11)),
blue(line(3,-2,9,6)), blue(line(3,-2,9,-10)), locate(9.3,2,"h=8"),
locate(9.3,-5.5,8),

red(arc(3,-2,20,-20,40,70)), red(arc(3,-2,20,-20,290,320)),
locate(9,6.5,"(9,6)"), locate(9,-10,"(9,-10)"), locate(6,-2,6),
locate(5,2.5,10), locate(5,-5.7,10)
)}}}

Edwin</pre>