a. Setting the right side to 0 and expanding each equation gives:

Equation [1]:

(x + 2)2 + (y − 3)2 = 25

(x + 2)2 + (y − 3)2 − 25 = 0

x2 + 4x + 4 + y2 − 6y + 9 − 25 = 0

x2 + 4x + y2 − 6y − 12 = 0 [3]

Equation [2]:

(x − 1)2 + (y + 4)2 = 16

(x − 1)2 + (y + 4)2 − 16 = 0

x2 − 2x + 1 + y2 + 8y + 16 − 16 = 0

x2 − 2x + y2 + 8y + 1 = 0 [4]

Solving the above 2 results simultaneously, we subtract Row [4] from Row [3] and this gives:

6x − 14y − 13 = 0

Solving for y gives:

`y=3/7x-13/14`

This means the intersection points are on the line

`y=3/7x-13/14`


b. Solve one of the circle equations for y using:

`y=(-b+-sqrt(b^2-4ac))/(2a)`

`y^2-6y+(x^2+4x-12)=0`

This gives:

`y=3+-sqrt(21-x^2-4x`


c. Substitute the positive case into the LHS of

`y=3/7x-13/14`, gives us:

`3+sqrt(21-x^2-4x)=3/7x-13/14`


d. Solve for x:

`sqrt(21-x^2-4x)` `=3/7x-13/14-3` `=3/7x-55/14`

Square both sides:

`21-x^2-4x` ` =(3/7x-55/14)^2` `=3025/196-165/49x+9/49x^2`

Re-writing for convenience:

`21-x^2-4x` `=3025/196-165/49x+9/49x^2`

Moving everything to the right hand side:

`0=9/49x^2+x^2-165/49x` `+4x` `+3025/196` `-21`

Simplifying:

`0=58/49x^2+31/49x-1091/196`

This gives us a quadratic in x

`58/49x^2+31/49x-1091/196=0`

or more simply

`58x^2+31x-272 3/4=0`

`x=-31/116+7/116sqrt1311` `=1.9177`

`x=-31/116-7/116sqrt1311` `=-2.4522`

e. Substitute these two x-values into

`y=3/7x-13/14`:

`[3/7x-13/14]_(x=1.9177)=-0.1067`

`[3/7x-13/14]_(x=-2.4522)=-1.9795`

So the points of intersection are (1.9177, −0.1067) and (−2.4522, −1.9795).

[We could have substituted the x-values into either circle equation and solved for y, but what I have done is easier.]

We can see that the circles, the line `y=3/7x-13/14` and the intersection points are all correct when we draw the graph:

12345-1-2-3-412-1-2-3-4xyOpen image in a new page

As a comment, this question could be solved very quickly, and easily, using a computer graphics program. We would just need to zoom in on the intersection points until we obtained the required precision.

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