Coordinate Lines and Circles
Find the intersection of the lines and .
Why can you not solve for the intersection of the lines and ?
Do the lines and have any intersection points?
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?
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?
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.
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?
Suppose a point has Cartesian coordinates (x,y). What are its polar coordinates? Suppose a point has polar coordinates . What are its Cartesian coordinates?
Consider the equation . I claim that this determines a circle. Which circle? (I.e. find the circle's center and radius.) (Hint: Factor )
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 c is 1 and then you get . 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 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 . If this equation is correct then -4 = ...?
Find the intersections between the circle and the line .
A very open-ended, frivolous and challenging question: How can you represent a line with an equation ... in three dimensions?