Expert: Paul Klarreich - 10/21/2008

Question
1. Find all possible solutions to the equations

x + y + 3z=1
3x - y + 2z=2
8x - 4y + 3z=5


2. The equation of a circle with centre C is (x-3)^2 +
(y-4)^2 =9 and O is the origin. The line joining O and C can be extended to meet the circle at P. Find the coordinates of P and show that the equation of the tangent to the circle at P is 3x + 4y=40.


Thanks very much for your help!  

Answer
Hi, Jamie,

You didn't indicate what you already tried to do, so I assume you just want an outline of the solutions.  If you have trouble, send me what you did and I'll try to get you over the stumbling block.

For your equations:

x + y + 3z   = 1  A
3x - y + 2z  = 2  B
8x - 4y + 3z = 5  C

You start by eliminating one of he variables.  I pick y, for that, because the first two have unit coefficients.

x + y + 3z   = 1  A
3x - y + 2z  = 2  B
-----------------------
4x + 5z = 3  (A+B)  D

4x + 4y + 12z   = 4  4A
8x - 4y + 3z    = 5   C
-------------------------
12x + 15z = 9 (4A+C)  E

Now you can combine D and E to get x and z, which you plug back into A,B, or C to get y.

................................
2. The equation of a circle with center C is
(x-3)^2 + (y-4)^2 = 9

and O is the origin.  The line joining O and C can be extended to meet the circle at P. Find the coordinates of P and show that the equation of the tangent to the circle at P is 3x + 4y = 40.

Carry out the following steps:

1. Show that the center C is (3,4).  

2. Show that the line passing through O and C has equation:

y = 4x/3,  or  3y - 4x = 0.

3. Find the point(s) of intersection of the line OC and the circle by solving the pair of equations

y = 4x/3
(x-3)^2 + (y-4)^2 = 9

Show that the solutions are the points.   P1(24/5, 32/5) P2(6/5, 8/5)

[Sorry about the fractions, but I didn't make this up.]

4. Since you are EXTENDING, P1 is the choice.

5. Look up the geometry theorem that says:  A tangent to a circle is ..... to the radius at the point of tangency.

6. Use that theorem to deduce the slope of the line tangent to the circle from the slope of OC.

7. Now that you have the slope and a point p1 on the line, use the point-slope form:

y = y1 = m(x - x1)

to complete the equation.