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I need to find the points of tangency on a circle (x^2+y^2=100) and a line y=5x+b the only thing I know about b is that it is negative. This line runs parallel to the line y=5x+7. I need to find the points of tangency between the line y=5x+b and the circle.

ian lindell
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  • What's the derivative of the equation representing a circle? What's the derivative of your line? – actinidia Jan 09 '18 at 18:12
  • https://math.stackexchange.com/questions/2254073/the-line-y-mxc-is-a-tangent-to-x2y2-a2-if https://math.stackexchange.com/questions/774250/finding-the-equations-of-the-lines-and-tangent-to-the-circle – lab bhattacharjee Jan 09 '18 at 18:17
  • Please, if you are ok, you can accept the answer and set it as solved. Thanks! – user Feb 04 '18 at 00:07

3 Answers3

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solve the equation $$(5x+b)^2+x^2-100=0$$ and set the discriminatin equal to Zero to find $b$

Dr. Sonnhard Graubner
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The distance from the center to the line is $$\frac {-b}{\sqrt {26}}=10$$

hamam_Abdallah
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The tangent will be perpendicular to the radius $y = -\frac 15 x.$

This line intersects the circle.

$x^2 + (-\frac 15 x)^2 = 100\\ \frac {26}{25} x^2 = 100\\ x = \pm 50\frac {\sqrt {26}}{26},y = \mp 10 \frac{\sqrt {26}}{26}$

There are two points of tangency.

$(y+10\frac {\sqrt{26}}{26}) = 5(x-50\frac {\sqrt{26}}{26})\\ (y-10\frac {\sqrt{26}}{26}) = 5(x+50\frac {\sqrt{26}}{26})$

Doug M
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  • The line y=5x+b can't have two points of tangency, the definition of tangent is that it touches at one point. – ian lindell Jan 09 '18 at 19:03
  • Sure it can. For two different values of $b, y = 5x + b_1$ and $y = 5x + b_2$ are parallel lines and each can be tangent to opposite sides of the circle. – Doug M Jan 09 '18 at 19:06
  • ok that makes sense, still if you solve for x and y you get, (9.8, -1.96) those aren't the correct answers. – ian lindell Jan 09 '18 at 19:09
  • @ianlindell Those are the correct coordinates of the tangent point for the problem as you’ve posed it here. If that’s not what your answer key has, then it’s either wrong (it happens), or you’ve misinterpreted the question. – amd Jan 09 '18 at 19:24
  • I would say $(25 \frac {\sqrt {26}}{13}, -5\frac {\sqrt{26}}{13}) \ne (9.8, -1.96)$ It is close, but it is not exact – Doug M Jan 09 '18 at 19:27