Find all points with integer coordinates that are a distance of square root of 29 from (-4,3)
how do u do this?????!?!?!?!
(x+4)^2 + (y-3)^2 = 29
So 29 has to be sum of two squares.
29 = 25 + 4
Those are the only two integers.
so ..
Case 1: (x+4)^2 = 25 and (y-3)^2 = 4
or
Case 2: (x+4)^2 = 4 and (y-3)^2 = 25
Case 1:
then x + 4 = ±2 and y-3 =±5
x = -2, y = 8
x = -6, y = 8
x = -2 , y = -2
x = -6 , y = -2
Case 1:
x+4 =±5 and y-3 = ±2
x = -1 , y = 5
x = -1 , y = 1
x = -9 , y = 5
x = -9 , y = 1
Case 2:
then x + 4 = ±2 and y-3 =±5
x = -2, y = 8
x = -6, y = 8
x = -2 , y = -2
x = -6 , y = -2
So it looks like 8 points.
(x+4)^2 + (y-3)^2 = 29
So 29 has to be sum of two squares.
29 = 25 + 4
Those are the only two integers.
so ..
Case 1: (x+4)^2 = 25 and (y-3)^2 = 4
or
Case 2: (x+4)^2 = 4 and (y-3)^2 = 25
Case 1:
x+4 =±5 and y-3 = ±2
x = 1 , y = 5
x = 1 , y = 1
x = -9 , y = 5
x = -9 , y = 1
Case 2:
then x + 4 = ±2 and y-3 =±5
x = -2, y = 8
x = -6, y = 8
x = -2 , y = -2
x = -6 , y = -2
So it looks like 8 points.
To find all points with integer coordinates that are a distance of the square root of 29 from (-4,3), we can use the distance formula.
The distance formula is given by:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, (-4,3) is the point (x1, y1), and the distance is the square root of 29.
So, substituting the values into the formula, we get:
sqrt(29) = sqrt((x2 - (-4))^2 + (y2 - 3)^2)
To simplify, we square both sides of the equation:
29 = (x2 - (-4))^2 + (y2 - 3)^2
Now, let's solve the equation step-by-step:
1. Expand the equation:
29 = (x2 + 4)^2 + (y2 - 3)^2
2. Since we are looking for integer coordinates, the terms (x2 + 4) and (y2 - 3) should be integers. Let's consider all possible combinations of these integers:
- When (x2 + 4) = 0, we have (y2 - 3)^2 = 29. This has no integer solutions since the square root of 29 is not an integer.
- When (x2 + 4) = ±1, we have (y2 - 3)^2 = 28. In this case, (y2 - 3) = ±√28. Simplifying further, we have (y2 - 3) = ±2√7. Solving for y2 gives:
- When (y2 - 3) = 2√7, we have y2 = 3 + 2√7.
- When (y2 - 3) = -2√7, we have y2 = 3 - 2√7.
- Similarly, you can substitute different values for (x2 + 4) and solve for y2 to get more points with integer coordinates.
So, the points with integer coordinates that are a distance of the square root of 29 from (-4,3) are:
1. (-4, 3 + 2√7)
2. (-4, 3 - 2√7)
3. (x2, 3 + 2√7) (where x2 is an integer)
4. (x2, 3 - 2√7) (where x2 is an integer)
Please note that there could be more points depending on the values of (x2 + 4) and (y2 - 3) that satisfy the equation.