Does anyone know how to solve this?

[DanielChris]

There is a circle with a radius of 2 where the center is at the origin, and a line 3x+4y−12=0 in the plane. The minimum distance between a point on the circle and a point on the line is?

 

[Dr.Peterson]

There is a circle with a radius of 2 where the center is at the origin, and a line 3x+4y−12=0 in the plane. The minimum distance between a point on the circle and a point on the line is?

I can think of several ways, depending on what you've learned. You may know a formula for the distance from a line to a point, using your equation; this is especially easy when the point is the origin.

If you don't know that, you could use the normal vector to the line to find the nearest point.

Or you could use various geometrical theorems. Or you could use calculus to minimize the distance to a point.

But all these ways involve the distance to the origin in some form; the important thing will be that you see how the distance from the origin is related to the minimum distance from the circle, which is based on simple geometry.

So, what have you learned that this exercise might be intended to exercise?

 

[DanielChris]

I can think of several ways, depending on what you've learned. You may know a formula for the distance from a line to a point, using your equation; this is especially easy when the point is the origin.

If you don't know that, you could use the normal vector to the line to find the nearest point.

Or you could use various geometrical theorems. Or you could use calculus to minimize the distance to a point.

But all these ways involve the distance to the origin in some form; the important thing will be that you see how the distance from the origin is related to the minimum distance from the circle, which is based on simple geometry.

So, what have you learned that this exercise might be intended to exercise?

I tried using the formula for the distance from a line to a point and got 3x+4y−12=0=>y=-(3/4)x+3 but I don't know what to do after this.

 

[pka]

There is a circle with a radius of 2 where the center is at the origin, and a line 3x+4y−12=0 in the plane. The minimum distance between a point on the circle and a point on the line is?

Can you write the equation of the line that contains the centre of the circle and is perpendicular to the given line?

 

[jonah2.0]

Beer soaked diagram follows.

There is a circle with a radius of 2 where the center is at the origin, and a line 3x+4y−12=0 in the plane. The minimum distance between a point on the circle and a point on the line is?

A picture is said to be worth a thousand words.
Can you paint the picture with equations?

 

[Dr.Peterson]

I tried using the formula for the distance from a line to a point and got 3x+4y−12=0=>y=-(3/4)x+3 but I don't know what to do after this.

That doesn't look like a formula for distance; you've just solved the equation of the line for y.

What is "the formula for the distance from a line to a point" that you have learned?

 

[pka]

I tried using the formula for the distance from a line to a point and got 3x+4y−12=0=>y=-(3/4)x+3 but I don't know what to do after this.

To DanielChris. I had hoped that you could respond to the several hints we gave you all containing the idea of a perpendicular distance.
If we want to find the distance from point \(P\) to the line \(\ell\). To do that we find the equation of the line which contains \(P\) and is perpendicular to \(\ell\), call it \(\ell_{\bot}\) Now suppose that \(\ell_{\bot}\cap\ell=\{Q\}\). So the distance from \(P\) to \(\ell\) is simply the length of \(\overline{PQ}\). Applied to this question: the line which contains the centre of the circle and perpendicular to the given line intersects the circle in two points but the given line in one point. How do we proceed