Math Help

Here is a Calculus probelm i have on one of my work sheets. I can do these when ther is a "_X" in it but this problem does not have one so im stuck. Any help please.

Factor & Solve

x^2 + y^2 - 18y - 4 = 0

(^ = to the power of)



Here is an example of one that i can do.

X^2 + 12X + y^2 - 8y = 5
(x^2 + 12X + _) + (y^2 - 8y + _) = 5 + _ + _
(x^2 + 12X + 16) + (y^2 - 8y + 9) = 5 + 16 + 9
(x+6)(x+6) + (y+4)(y+4) = 57
(x+6)^2 + (y+4)^2 = sqRoot 57

Center = 4,3 Radius = 3sqRoot3

Please help!

Thanks



cooked

 

First, your example is incorrect. Note that it is -8y and not +8y and thus it should be (y-4)^2.

If (x+6)(x+6) + (y+4)(y+4) is 57 then this (x+6)^2 + (y+4)^2 cannot be the square root of 57.

Answer to example is thus: (x+6)^2 + (y-4)^2 = 57. This being the equation of a circle with centre at (-6, 4) and a radius of sqRoot 57 = 7.5498... Note this is not 3sqRoot3.

For details of equations of a circle see: http://www.analyzemath.com/CircleEq/CircleEq.html

The answer to your original problem is: a circle with centre at (0, 9) and a radius of sqRoot 85 = 9.2195...

The method to solving these in general is:

1) Problem is factor and solve:

x^2 + ax + y^2 + by = c

where a, b and c are numbers.


2) Factorising x and y:

(x + p)^2 would give you x^2 + 2px + p^2
(y + q)^2 would give you y^2 + 2qy + q^2

where p and q are numbers.


3) Substituting the factors:

a = 2p
b = 2q
and right hand side is c + p^2 + q^2.

thus:
x^2 + ax + p^2 + y^2 + by + q^2 = c + p^2 + q^2

hence:
(x + p)^2 + (y + q)^2 = c + p^2 + q^2


4) Convert to circle dimensions:

Radius = sqRoot ( c + p^2 + q^2 )
Centre is at (-p, -q).


In your given problem: a=0, b=-18 and c=4 thus (from step 3) p=0 and q=-9. Plugging in these values of c, p and q in step 4 gives you your answer.

You can use this method for solving all such equations provided you stick to the real domain and don't end up with complex numbers as a result of a square root of a negative number.
---------------
Give a man a fish he feeds for a day.
Teach a man to fish he feeds for life.
Chinese proverb.

 

Thanks Maxwell! That is just what i needed! I was able to work through the problems with my teacher, and she pointed out what i was doing wrong, and steered me in the right direction.

Thanks



cooked