Finding the Equation of a Circumference with a Tangent Line and Center at (0,0)

[FrostScYthe]

What's the equation of a circumference that has center on (0,0) and a tangent line x + 3y = 10?

The answer is supposed to be x^2 + y^2 = 10, but I don't know how to get there.

 

[VietDao29]

If you have a circle with center O (0, 0), and a tangent line x + 3y = 10. Can you find out the distance between the center, and the tangent line? Is it also the radius of the circle?
Having the center O (0, 0), and its radius, can you find the equation of that circle? :)

 

[FrostScYthe]

Very helpful

but there is more than 1 distance from (0,0) to the x + 3y = 10 line, how do I know which distance will give me a radius for a circle that will only touch the line at 1 point?

 

[VietDao29]

but there is more than 1 distance from (0,0) to the x + 3y = 10 line, how do I know which distance will give me a radius for a circle that will only touch the line at 1 point?

What do you mean by 'more' than 1 distance? The distance between a point and a line is, roughly speaking, the shortest line segment, that have one endpoint is that point, and the other point lies on the line.
Say, you have a point M(x_M, y_M), and the line $$\Delta$$, whose equation is: ax + by + c = 0.
Do you know the formula:
$$d ( M, \ \Delta) = \frac{|ax_m + by_m + c|}{\sqrt{a ^ 2 + b ^ 2}}$$?

 

[HallsofIvy]

but there is more than 1 distance from (0,0) to the x + 3y = 10 line, how do I know which distance will give me a radius for a circle that will only touch the line at 1 point?

No, there is only one distance from a point to a line- it is the shortest distance from that point to any point on the line. It should take only a little thought to see that dropping a perpendicular to the line from the point will give the shortest distance (think about the fact that the hypotenuse of a right triangle is the longest side).

What is the slope of the line x+ 3y= 10?
What is the slope of a line perpendicular to that?
What is the equation of the line through (0,0) with that slope?
Where does that line intersect the line x+ 3y= 10?

 

[FrostScYthe]

So then the perpendicular line for y = (10 - x)/3, which has a slope of
-x/3 would be 3x and then the equation has to go through (0,0) and y = 3x goes through zero, and the intersection between the two lines would (1,3) hmm which then we measure the distance

sqrt((3-0)^2 + (1-0)^2) = sqrt(10) = distance = radius

and then the circle would be x^2 + y^2 = 10 ...!

Thanks so much Ivy and to the others too =)