SOLUTION: Pleas help me find the exact values of m for which the line y=mx+5 is a tangent to the curve x^2+y^2=10. Your help will be much appreciated. :D

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Question 1089590: Pleas help me find the exact values of m for which the line y=mx+5 is a tangent to the curve x^2+y^2=10.
Your help will be much appreciated. :D
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Substitute y = mx+5 into the circle equation. You will get

x%5E2+%2B+%28mx%2B5%29%5E2 = 10,

x%5E2+%2B+m%5E2x%5E2+%2B+10mx+%2B+25 = 10,

%281%2Bm%5E2%29x%5E2+%2B+10mx+%2B+10 = 0.             (1)


The condition that the straight line is tangent to the circle is equivalent to the fact that the equation (1) has only one root,

which, in turn, means that the discriminant of the equation (1) is equal to zero.


It gives you an equation for m:

d = %2810m%29%5E2+-+4%2A10%2A%281%2Bm%5E2%29 = 0.


Simplify and solve it for m:

100m%5E2+-+40m%5E2+-+40 = 0,

60m%5E2 = 40  ====>  m%5E2 = 40%2F60 = 2%2F3  ====>  m = +/- sqrt%282%2F3%29.

Solved.

The lesson to learn from this solution: 

    The fact that a line is tangent to the circle is equivalent to the fact that they have only one common point.