Find all points (x,y) that are 13 units away from the point (2,7) and that lie on the line x - 2y = 10. Give your answer as a list of points separated by semicolons, with the points ordered such that their x-coordinates are in increasing order. So "(1,-3); (2,3); (5,-7)" - without the quotes - is a valid answer format.)

Question

Find all points (x,y) that are 13 units away from the point (2,7) and that lie on the line x - 2y = 10. Give your answer as a list of points separated by semicolons, with the points ordered such that their x-coordinates are in increasing order. So "(1,-3); (2,3); (5,-7)" - without the quotes - is a valid answer format.)

Answer 1

all points (x,y) that are 13 units away from the point (2,7) defines a circle:

(x-2)^2 + (y-7)^2 = 169

so, the line x-2y=10 may intersect the circle in 0,1, or 2 points.

So, plug x=2y+10 into the equation of the circle and see where the two graphs intersect.

Answer 2

see whether you get what wolframalpha does:

https://www.wolframalpha.com/input/?i=solve+%28x-2%29^2+%2B+%28y-7%29^2+%3D+169,+x-2y%3D10

Answer 3

To find all the points (x, y) that are 13 units away from the point (2, 7) and that lie on the line x - 2y = 10, we can follow these steps:

Step 1: Find the general equation of the circle with a center at (2, 7) and a radius of 13 units.
The general equation of a circle centered at (h, k) with radius r can be given as:
(x - h)^2 + (y - k)^2 = r^2

Plugging in the values from the given information, we have:
(x - 2)^2 + (y - 7)^2 = 13^2
Simplifying, we get: (x - 2)^2 + (y - 7)^2 = 169

Step 2: Rewrite the given line equation in terms of y to express it as a function of x.
x - 2y = 10
Solving for y, we have: y = (x - 10) / 2

Step 3: Substitute the expression for y from Step 2 into the equation from Step 1.
(x - 2)^2 + ((x - 10) / 2 - 7)^2 = 169
Simplifying and expanding, we get:
(x - 2)^2 + ((x - 10) / 2 - 7)^2 = 169

Step 4: Solve the quadratic equation in Step 3 to find the values of x.
Expanding and solving the quadratic equation, we get the solutions x = -15 and x = 5.

Step 5: Substitute the values of x into the equation from Step 2 to find the corresponding y-values.
For x = -15, y = (x - 10) / 2 = (-15 - 10) / 2 = -25 / 2 = -12.5
For x = 5, y = (x - 10) / 2 = (5 - 10) / 2 = -5 / 2 = -2.5

So, we have two points, (-15, -12.5) and (5, -2.5), that satisfy the given conditions and lie on the line x - 2y = 10. These points are 13 units away from the point (2, 7).

Therefore, the answer is: (-15, -12.5); (5, -2.5)