Expert: Scott A Wilson - 5/28/2009

Question
1.  What is the most possible number of solutions of one circle and a straight line in the same system?

2.  Find the exact solutions of
x^2 - (y - 6)^2 = 36 and y = -x^2

3.  Solve the system, 4x^2 + y^2 = 20 and y = 4x

4.  In which quadrant is the terminal side of a 455° angle?

5.  In which quadrant is the terminal side of a -135° angle?

6.  Convert -270° to radian measure.

7.  Write the equation in standard form y = x^2 - 10x + 4.

8.   Find the midpoint of the line between (2, 8) and (4, 6).

9.  Find the center and radius of the circle with the given equation
x^2 + y^2 + 6y = -50 - 14x.

Thank you so much for your time :)


Answer
1.  What is the most possible number of solutions of one circle and a straight line in the same system?

A line and a circle can cross at one point,
and that is where the line can go inside the circle.
The line can go out at another point,
so that is the second time it crosses,
at which point it leaves the circle.
It looks like the biggest number is two.

2.  Find the exact solutions of
x^2 - (y - 6)^2 = 36 and y = -x^2

The equation is x² - y² + 12y - 36 = 36.
Subtract 36 from both sides and you get x² - y² + 12y - 72 = 0.
If y = -x², then we have -y - y² + 12y - 72 = 0.
Combining the y's gives -y² + 11y - 72 = 0.
Changing the signs gives y² - 11y + 72 = 0.
Using the quadratic formula, note that there is a
√(b² - 4ac) in the formula where a=1, b=-11, and c=72.
This is √(121 - 288) = √-167.
So we could have (11±(√167)i)/2, if i is allowed as i = √(-1)

3.  Solve the system, 4x^2 + y^2 = 20 and y = 4x

Putting in y = 4x gives us 4x² + 16x² = 20.
That means that 20x² = 20, or x² = 1.  The answer would be x = ±1.


4.  In which quadrant is the terminal side of a 455° angle?

Most angles are known to be between 0° and 360°,
and past that, 360° should be subtracted
since 360° is just a circle.
So 455° - 360° = 95°.  Quadrant I is 0° to 90°.
Since this is 5° past 90°, the angle is in quadrant II.


5.  In which quadrant is the terminal side of a -135° angle?

To get this between 0° and 360°, add 360° to it.
-135° + 360° = 225°.  Quadrant III is 180° to 270°.
 225° is in between 180° and 270°.


6.  Convert -270° to radian measure.
   To convert to radians, multiply by π/180.


7.  Write the equation in standard form y = x^2 - 10x + 4.

Standard from is y-k = (x-h)².
To complete the square, take half of 10 and square it.
Half of 10 is 5.  Square 5 and the answer is 25.

Since there is 4 on the x side already,
all that needs to done to get to 25 is add 21 to both sides.

The answer is then y+21 = (x-5)², so h = 5 and k = -21.


8.   Find the midpoint of the line between (2, 8) and (4, 6).

The points are (x1, y1) = (2, 8) and (x2, y2) = (4, 6).
The midpoint is ((x1+x2)/2, (y1+y2)/2).


9.  Find the center and radius of the circle with the given equation
x^2 + y^2 + 6y = -50 - 14x.

Convert to the form (x-a)² + (y-b)² = r².
The center is at (a, b) and the radius is r.
We have x² + 14x + y² + 6y =50.
To complete the squares, add (14/2)² + (6/2)² to both sides.
The answer is (x + 7)² + (y + 3)² = 50 + 49 + 9.
Adding up the 50 and 49 gives 99,
and then adding on 9 more gives 108.
This makes r √108.
From this information, the standard from can be gotten.


 Thanks for the questions.  I'm enjoy answering them :·)