Find all points (x,y) that are 13 units away from the point (2,7) and that lie on the line x-2y=10.
symbolize and solve equations using:
average prices of three items A, B, C is $ 16000, if the price of the item a is twice the sum of the prices of the items B and
symbolize and solve equations using:
average prices of three items A, B, C is $ 16000, if the price of the item a is twice the sum of the prices of the items B and C
√((x-2)^2 + (y-7)^2 = 13
square both sides and expand
x^2 - 4x + 4 + y^2 - 14y + 49 = 169
x^2 - 4x + y^2 - 14y = 116
but x = 10+2y
sub that into the first equation
(10+2y)^2 - 4(10+2y) + y^2 - 14y = 116
100 + 40y + 4y^2 - 40 -8y + y^2 - 14y = 116
5y^2 + 18y - 56 =0
(y-2)(5y + 28) = 0
y = 2 or y = -28/5
if y = 2, then x = 10+2(2) = 14 ---> point is (14,2)
if y = -28/5, x = 10 + 2(-28/5) = -6/5 ---> point is (-6/5 , -28/5)
To find the points (x, y) that are 13 units away from the point (2, 7) and lie on the line x - 2y = 10, we can use the distance formula and substitution. Here's how you can do it:
Step 1: Distance formula
The distance formula between two points (x1, y1) and (x2, y2) is given by:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In our case, the distance is 13 units, and the points are (2, 7) and (x, y). So, we have:
13 = sqrt((x - 2)^2 + (y - 7)^2)
Step 2: Squaring both sides
To eliminate the square root, we square both sides of the equation:
13^2 = (x - 2)^2 + (y - 7)^2
169 = (x - 2)^2 + (y - 7)^2
Step 3: Substitution
Now, we substitute the value of x from the equation of the line x - 2y = 10 into the equation we obtained in step 2:
x - 2y = 10
x = 2y + 10
169 = (2y + 10 - 2)^2 + (y - 7)^2
169 = (2y + 8)^2 + (y - 7)^2
Step 4: Solving for y
Expand and simplify the equation:
169 = 4y^2 + 32y + 64 + y^2 - 14y + 49
169 = 5y^2 + 18y + 113
Subtract 169 from both sides:
5y^2 + 18y + 113 - 169 = 0
5y^2 + 18y - 56 = 0
Step 5: Solve the quadratic equation
Now we solve the quadratic equation using factoring, completing the square, or using the quadratic formula. In this case, the equation factors easily:
(5y + 14)(y - 4) = 0
So, we have two possible values for y:
1. 5y + 14 = 0 => y = -14/5
2. y - 4 = 0 => y = 4
Step 6: Substitute y values and solve for x
For each value of y, substitute it back into the equation x = 2y + 10 to find the corresponding x values:
For y = -14/5:
x = 2(-14/5) + 10
x = -28/5 + 50/5
x = 22/5
For y = 4:
x = 2(4) + 10
x = 8 + 10
x = 18
So, the two points (x, y) that are 13 units away from (2, 7) and lie on the line x - 2y = 10 are:
(22/5, -14/5) and (18, 4).