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Featured 1 month ago

A circle with a radius

Recall that the distance between a point

Given that the circle also touches the line

Hence, the largest radius

Featured 1 month ago

Transforming the given equation of the circle

So the center of the circle C

and radius of the circle

If

So

Now

So length of each tangent

Now

So angle between the tangents will be

But

Featured 4 weeks ago

Well now I have to look up what the double quote means. Probably minutes or seconds.

OK, Wikipedia says **For example, 40.1875Â° = 40Â° 11' 15"** .

That's 40 degrees, 11 minutes, 15 seconds, a minute being 1/60th of a degree and a second being 1/60th of a minute. (Thank you Babylonians.)

So we're to understand 1920" as

So we're looking at a diameter

Check: Google

Featured 5 days ago

The general equation of a circle is:

Where:

Circle A

If B is translated by

Centre:

Circle B:

By using the distance between the centres and the radii we can deduce the following:

Let:

If:

Using the distance formula:

Sum of radii:

So the circles intersect at two points or one is contained in the other. This can be tested by noticing that if the diameter of the smaller circle is less than the radius of the larger then the smaller circle is contained in the larger one.

Diameter of smaller circle is

Radius of larger circle is

So smaller circle is contained in the larger.

To find the shortest distance:

PLOT:

Featured 2 days ago

We're oddly given a circle in this problem and awkward wording asking for the smallest possible x coordinate.

I'm not sure what the circle has to do with anything. We could list equations endlessly for circles containing these two points. Forget about the circle.

Given one side, there will be two equilateral triangles with that as a side. Let's rewrite the problem:

Find all possible third vertices

There are a few different ways to do this. I'd lean toward complex numbers, but I probably should try to keep this simpler than that.

Let's review. The altitude

We just need to go

The midpoint D of AB is

The direction vector from A to B is

For the perpendicular direction vector we swap and negate one:

Now schematically what we're doing is

We have

That's the general solution; let's apply it to

Uh, the one with the least x coordinate is

Check:

We check the squared distance to each point

Featured yesterday

See below.

This question is a little ambiguous. I'm guessing you mean a volume of revolution. If this is the case then a rectangle rotated about a line could be a solid cylinder or a hollow cylinder ( Tube) if the line of rotation is parallel or perpendicular to the sides of the rectangle. If the line of rotation is neither parallel nor perpendicular then a vast number of different solids could result. A big problem here is on the definition. Strictly a prism has a base that is a polygon and faces that are flat, but you could say a cylinder has a polygonal base with an in finite number of sides and an infinite number of faces.

Here are two rotations of a rectangle around different lines.

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