find the eqn of the circle that passes through the point A(8,1) and B(7,1) and has , for its tangent at B, the line 3x-4y-21=0

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- Jul 3rd 2009, 02:29 AMhelloyingeqn of circle and tangent
find the eqn of the circle that passes through the point A(8,1) and B(7,1) and has , for its tangent at B, the line 3x-4y-21=0

- Jul 3rd 2009, 02:53 AMProve It
The equation of a circle is given by

, where is the centre and is the radius.

From the information given, we can generate three equations in three unknowns, so that we can solve them simultaneously for and .

Substituting point A into the equation gives

.

Substituting point B into the equation gives

.

This circle's derivative is given by:

.

This is the derivative, which gives us the gradient of the tangent at all points on the circle.

If it's tangent at B is the line , we can rearrange this to read .

So the gradient of the tangent at point B is .

So at point the gradient is .

Substituting these values into the derivative gives

.

We can now solve for and .

So far we have:

and .

Therefore

.

We also know:

, so

.

Finally, we know that

.

Now we finally have enough information for the equation of the circle:

. - Jul 3rd 2009, 04:55 AMTesla23
centre of circle must lie on perp bisector of AB which is line x = 7.5, so let centre be C be the point (7.5,y)

so we need to find y such that |AC| = distance of C to line 3x-4y-21=0 so:

which you can solve for y

Edit: I missed the "Tangent at B" part of the question, and as pointed out by HallsOfIvy the line 3x-4y-21=0 does not pass through B, this solution finds the circle passing through A and B with tangent 3x-4y-21=0 - Jul 3rd 2009, 05:05 AMHallsofIvy
Actually, the condition given, "passes through the point A(8,1) and B(7,1) and has , for its tangent at B, the line 3x-4y-21=0", is impossible. 3(7)- 4(1)- 21= 21- 4- 21= -4, not 0 so the line 3x- 4y-21= 0 does not even pass through B.

Tesla23's solution actually gives the circle passing through A and B and tangent to 3x- 4y- 21= 0 but not]**at**B. - Jul 3rd 2009, 06:05 AMhelloying
- Jul 3rd 2009, 06:24 AMProve It