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## High school geometry

### Course: High school geometry>Unit 6

Lesson 3: Problem solving with distance on the coordinate plane

# Challenge problem: Points on two circles

Watch Sal solve a challenging problem where he has to determine if points are on both, one, or neither of two circles. Created by Sal Khan.

## Want to join the conversation?

• At ,It is mentioned how we find the (Ra)2.How do we do it i don't get the (-5-0)2
• it is straight up the Pythagorean theorem we can derived a formula which is what we call the distance formula: d=sqrt[(x_2-x_1)^2 + (y_2-y_1)^2] (*We used distance formula iff we're given two points to identify the displacement of those points) . In sal example, he wants to find the radius between center A and point P after using the Pythagorean theorem (or distance formula) he conclude that the radius is 5*sqrt(2).
• Would D not be considered to be "In' the circle, why was it classified as "Neither?"
• The question was whether the point is on circle A, circle B, or neither. We don't care if it's inside or outside of either circle.
• Why not draw the 2 circle and place the points and see if these are on circle A and B ?
• Sure, a graphical solution is one way of solving the problem - and in this case, it might actually be easier. When you're not dealing with integers, though - for example, if you want to know if the point P(e, pi) lies on some circle - you're going to have to do that using the distance formula, so it's best to learn the method.

He kind of does that in this video for points D and E, but it's probably best to do it properly with compasses and grid paper.
• Wait, I understand all the math that he used to get to the solutions, however, when it says "Point P is ON circles A and B" how do we know that it is on the perimeter of the circle? When a point is on a circle, can't it be anywhere on the circle, therefore deeming this problem unsolvable??
• The definition of a circle in analytic geometry is: the set of all points located the same distance away from a given point and is algebraically defined by the equation

``(x-h)^2 + (y-k)^2 = r^2``

where (h,k) is the center point and r is the radius. There is no magic equation, as far as I know, that defines all points on the circumference and inside a circle, so in your interpretation of a circle there is no equation to solve that problem. You could, however, turn the above equation into an inequality to determine if "Point P" `(x,y)` is a certain distance `(r)` from the center of the circle `(h,k)`.

``ex. if: (x-h)^2 + (y-k)^2 <= r^2then it is on the circle``
• What's a locus? I've read the dictionary definition but what does it encompass? What can you use it to describe?
• looks like...'locus' means all the answers to a kind of problem that has a group of solutions instead of just one exact one.
the word is latin-ish for "this is where you'll find it"
• ....Is Sal assuming that it is a right angle?
• No. It's a coordinate plane
(1 vote)
• While solving point C, why is the distance formula only applied with reference to point B and not A or P?
• Sal took the sqrt8 and turned it into 2*sqrt2 ,so why didn't he take sqrt10 and simplify it to 5*sqrt2?
(1 vote)
• √8=√(4•2)=√4•√2=2√2
√10=√(5•2)=√5•√2. There are no perfect squares to work with in the factorization of 10.
• How do you know that the circles don't overlap and that point P (0,0) is in both of the circles? Would you know because the question says that point P is on circles A and B and doesn't say that point P is in circles A and B?