Question 1056900: Find the value/s of k for which the circle x^2+y^2=4 and (x-4)^2 + (y-3)^2=k^2 intersect at exactly one point.
Found 3 solutions by josgarithmetic, ikleyn, MathTherapy:
Answer by josgarithmetic(39620)   (Show Source):
You can put this solution on YOUR website! Begin making a graph according to this description to help see what you may be able to do:
Circle centered at origin (0,0) and radius 2;
Point for other circle center at (4,3), but radius is unknown, k.
A segment connects the two center points (0,0) and (4,3). The equation for the LINE for this segment is  .
THINK about that, and the drawing you hopefully made.
Do you imagine a process to continue and find your k?
As a big hint toward that, What is the intersection point for and  ?
What is the distance between that found intersection point and (4,3)?
That would be k.
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I did not solve this for you but explained precisely how you can solve it.
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Solving the nonlinear system  , omitting the steps for it here, should give you the intersection point, ( 8/5, 6/5 ). Use the distance formula to find distance from ( 8/5, 6/5) to (4,3). That is your k.
Answer by ikleyn(52803)  (Show Source):
You can put this solution on YOUR website! .
Find the value/s of k for which the circle x^2+y^2=4 and (x-4)^2 + (y-3)^2=k^2 intersect at exactly one point.
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There are two answers: k = 3 and k = 7.
Solution
1. x^2+y^2=4 is the circle of the radius 2 with the center at the origin (0,0).
2. (x-4)^2 + (y-3)^2=k^2 is the circle of the radius "k" with the center at the point (4,3).
Notice that this center lies at the distance 5 =  from the origin.
3. The condition requires that the circle #1 touches the circle #2.
It is possible in two cases:
k = 5 - 2 = 3 (exterior touching), and'
k = 5 + 2 = 7 (interior touching).
Solved.
Answer by MathTherapy(10552)  (Show Source):
You can put this solution on YOUR website! Find the value/s of k for which the circle x^2+y^2=4 and (x-4)^2 + (y-3)^2=k^2 intersect at exactly one point.
Since the circles INTERSECT at just 1 point, it obviously means that they only TOUCH each other at a point on the OUTSIDE of their circumferences
The 1st circle: [ ] is centered at the origin (0, 0), and has a radius of r = 2 ( )
The 2nd circle: [ ] is centered at (4, 3) and has a radius of r = k
Since the circles touch each other at 1 point, and the 2nd circle’s center is (4, 3), it follows that they MUST TOUCH each other in the 1st quadrant
When joined, the centers of the 2 circles form the 2 radii of the circles, and these radii, combined, happen to form the hypotenuse of a 3-4-5 right triangle. This hypotenuse = 5
Since the circle centered at (0, 0) has a radius of 2, it follows that the circle that’s centered at (4, 3) has a radius of 5 – 2, or 3, so 
The 2nd circle: [], centered at (4, 3), can also be much larger than the circle centered at (0, 0), and so, can have a  (3 + 2 + 2), or 3,
plus the diameter of the circle centered at (0, 0). This means that the smaller circle [centered at (0, 0)] will be inscribed in the larger circle.
If it's confusing, you'd see a much clearer picture if you draw a diagram, as I did.
It's as easy as that...nothing COMPLEX!
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