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 Question from Mahesh, a parent: find the length of the cord intercepted by a circle with equation X2 +Y2 -6X+4Y-12=0 with a line of equation 4X-3Y+2=0 Pl help me solve this problem thanks MNK

Hi Mahesh.

To solve this problem, I would proceed as follows:

1. Solve the line equation for Y.

2. Substitute the expression for Y (that you found in step 1) in for the variable Y in the standard circle equation. This gives you a quadratic with just the variable X in it. Simplify.

3. Solve the quadratic for X (by completing the square or factoring or using the quadratic formula: whichever you are comfortable with). You should get two values, corresponding to the two X values of the intersection points of the line and the circle. Use these determine the corresponding Y values by using the line equation in the Y = form.

4. Now you have the co-ordinates of the two points, so use the distance Formula to find the length of the chord.

Here is an example of how to solve this problem.

Question.

Find the length of the chord through the circle x2 + y2 + 6x + 4y - 3 = 0 which is created by the line x - y - 3 = 0.

Step 1: Solve the line for y:

y = x - 3

Step 2: Substitute this into the circle equation.

x2 + y2 + 6x + 4y - 3 = 0
x2 + (x - 3)2 + 6x + 4(x - 3) - 3 = 0

and simplify:
x2 + (x2 - 6x + 9) + 6x + 4x - 12 - 3 = 0
2x2 + 4x - 6 = 0
x2 + 2x - 3 = 0

Step 3: Solve the quadratic for x by completing the square:

(x2 + 2x + 1) - 3 = 1
(x + 1)2 = 4
x + 1 = +/- 2
x = {-3 , 1}

and find the corresponding values of y using the line equation from step 1:

y = -3 - 3 = -6 ... this means the point is (-3, -6).

y = 1 - 3 = -2 ... this means the point is (1, -2).

Step 4: Use the distance formula to find the distance between these points:

d = sqrt( (x2 - x1)2 + (y2 - y1)2 )
d = sqrt( (1 + 3)2 + (-2 + 6)2 )
d = sqrt( 42 + 42 )
d = sqrt( 32 )
d = 4 sqrt(2) = about 5.66 units.

Now you try your own question.

Cheers,
Stephen La Rocque.

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