find the standard for of the equation of the circle cntered at (2.-2) and touching the line (3x-2y=6) Find the circumference

Question

akashdeepdeb · Answer

So if we consider the radius to be a proper line then we have one point on the radius 
(x1,y1) =  (2,-2) and we have the slope of the radius which is m = -2/3
There is a point slope equation which we can use: y – y1 = m(x – x1)
when we substitute in equation we get the equation of the radius as 3y + 2x = -2
Now we have to find the point on the circle which we can because we know the 2 equations - 1 of the tangent and 1 of the radius!
When you solve both the equations you will get a point!
Then you will have 2 points the center and one point on the circle[(2,-2)]
Apply distance formula and you will get the distance between the 2 points ie. THE RADIUS and then put it in the circle equation = (x-x1)^2 + (y-y1)^2 = r^2
aNd then you will have the EQUATION OF THE CIRCLE!
Hope this helped!

anonymous · Answer

|dw:1376394011795:dw|Simple recipe to follow:

1. realise that the equation for a circle (x-a)^2 + (y-b)^2 = R^2 holds (with a,b is the centre of the circle, R is the (unknown) radius.

2. realise that there exists a line perpendicular to your original line, which goes through the centre of the circle.

3. rewrite the line to y = 3/2 * x - 3, so you can find the direction of the perpendicular line via -1 / (3/2) = - 2/3, so this perpendicular line looks like y = -2/3 * x + b

4. as this perpendicular line passes through the centre of the circle, you can use (2, -2) to calculate b from step 3. This gives you the full equation of the line.

5. calculate the intersection between the two lines (2 equations with 2 unknowns, so solvable). This intersection lies on your circle as you can see in the drawing.

6. determine the distance between the intersection on the circle and the centre of the circle, which equals the radius R that you are looking for.

7. Now determine the equation of the circle (see step 1) where you know a, b and R.

8. Find the circumference of the circle now you have the radius R.

Solved :-).