# Coordinate Lines and Circles

- Author:
- Adam

- Topic:
- Circle

## Problem 1.

Find the intersection of the lines and .

## Problem 2.

Why can you not solve for the intersection of the lines and ?

## Problem 3.

Do the lines and have any intersection points?

## Problem 4.

Suppose the lines and are perpendicular. In that case, solve for

*a*in terms of*b, c,*and*d*. Why don't*c*and*f*show up in the equation?## Problem 5.

You might think that, if you have the line in standard form, then the following is a valid way to find the slope of the line:
From this you can use the formula for the slope:
This is valid in most cases. However, it is not valid in one case. What is that case?

## Problem 6.

In the diagram below, vertical and horizontal lines intersect pair-wise in four points,

*C, D, E*and*F*. Point*E*has coordinates (*w,x*) and point*F*has coordinates (*y,z*). Find*w, x, y,*and*z*.## Problem 7.

The way that Cartesian coordinates work is: Tell me how far to go left-right (the

*x-*coordinate) and tell me how far to go up-down (the*y*-coordinate) and I can find your point. There is another way to specify points though, called polar coordinates. Let's see its use by an example. Let's start by extending a point from the origin along the*x*-axis by 2 units.Next let's rotate by an angle of 45-degrees.

This process has just landed at the point

*D*. In fact we can land on every point in the plane by giving similar instructions. If we had rotated by 90-degrees this would have landed on the point given in Cartesian coordinates as (0,2). We've just seen that in polar coordinates the point*D*has coordinates (2,45) for the "radius" of 2 and the "angle" of 45-degrees. What are the coordinates of point*D*above in Cartesian coordinates? Also, consider the point (-1,0) given in Cartesian coordinates. What are its coordinates if we instead use polar coordinates?## Problem 8.

Suppose a point has Cartesian coordinates ( . What are its Cartesian coordinates?

*x*,*y*). What are its polar coordinates? Suppose a point has polar coordinates## Problem 9.

Consider the equation . I claim that this determines a circle. Which circle?
(I.e. find the circle's center and radius.)
(Hint: Factor )

## Problem 10.

The previous problem was set up to be easy. If I had written the equivalent equation it would have been much less clear how you could write this as a circle equation. What we need is to be able to take an expression like and find a number so that when we add it on we get and then after factoring this turns into a perfect square . We've already seen that in this particular case the right choice for .
What about for ? What number could we add to this so that when we factor the expression, the result is a perfect square?
Hint: Think about . This equation essentially states what we want: a number . If this equation is correct then -4 = ...?

*c*is 1 and then you get*c*so that after factoring we get a perfect square. If this equation is correct then we can multiply out the square and get the new equation## Problem 11.

Find the intersections between the circle and the line .

## Problem 12.

A very open-ended, frivolous and challenging question:
How can you represent a line with an equation ... in

*three*dimensions?