find (x,y)--Precalculus

[danimalswim]

Find all points with -x+2y=0 so that (x,y) is 5 units from (-1,4). Everything I tried doesn't work. First, I tried solving -x+2y=0 with y equaling 0, 1, 2, 3, -1, -2, -3. When y=0 x=0, y=1 x=2, y=2 x=4, y=3 x=6, y=-1 x=-2, y=-2 x=-4, y=-3 x=-6. Then, I used the distance formula on all of these to see if any were 5 units from the point (-1,4). None of them worked. Please help me!!!

 

[Deleted member 4993]

Find all points with -x+2y=0 so that (x,y) is 5 units from (-1,4). Everything I tried doesn't work. First, I tried solving -x+2y=0 with y equaling 0, 1, 2, 3, -1, -2, -3. When y=0 x=0, y=1 x=2, y=2 x=4, y=3 x=6, y=-1 x=-2, y=-2 x=-4, y=-3 x=-6. Then, I used the distance formula on all of these to see if any were 5 units from the point (-1,4). None of them worked. Please help me!!!

Hint:

Equation of a circle of radius = r and centered at (h,k):

r² = (x+h)² + (y+k)²


Now find points of intersection of your line whose equation is

-x + 2y = 0

with the circle whose equation is:

5² = (x-1)² + (y+4)²

 

[JeffM]

Find all points with -x+2y=0 so that (x,y) is 5 units from (-1,4). Everything I tried doesn't work. First, I tried solving -x+2y=0 with y equaling 0, 1, 2, 3, -1, -2, -3. When y=0 x=0, y=1 x=2, y=2 x=4, y=3 x=6, y=-1 x=-2, y=-2 x=-4, y=-3 x=-6. Then, I used the distance formula on all of these to see if any were 5 units from the point (-1,4). None of them worked. Please help me!!!

Trial and error are not going to solve things for you in calculus, which builds on algebra. Why would you think that the answer must be an integer?

What frequently helps is to sketch a graph so that you can visualize what the problem involves. If you did so, you would see that
- x + 2y = 0 represents a straight line running through the origin and that (- 1, 4) is not on the line. What kind of figure do you get when you sketch all the points that are five units from a point. You get a circle, right? And a circle intersects a line in 0, 1, or 2 places.

The unknowns in your problem are the coordinates of the point or points of intersection. So define the unknowns briefly and assign a letter to each.

\(\displaystyle Let\ a\ be\ the\ x\ coordinate\ of\ the\ point\ of\ intersection\ between\ the\ line\ and\ the\ circle.\)

\(\displaystyle Let\ b\ be\ the\ y\ coordinate\ of\ the\ point\ of\ intersection\ between\ the\ line\ and\ the\ circle.\)

OK. You have two unknowns. To find them you will need two equations, right? You know that (a, b) is on the line. You know that (a, b) is 5 units away from (-1, 4). Can you express those two facts as mathematical equations? What are they? Can you solve them?