This may not be the most direct way......(someone else may nave a better idea)...well, anyway...
We have x - 2y = 10 or 2y = x - 10 or y = (1/2)x - 5
And a line perpendicular to the given one has a slope of -2
So....using (2, 7) and m = -2, we have
y = 7 = -2(x - 2)
y - 7 = -2x + 4
y = -2x + 11
And setting -2x + 11 = (1/2)x - 5 we have
-4x + 22 = x - 10 solve for x
5x = 32 → x = 32/5 = 6.4 and this is the x coordinate of the intersection of both lines
So ...the y intersection point of these two lines is -2(32/5) + 11 = -64/5 + 11 = -1.8
Now....we need to calculate the distance between this point and (2, 7)....it it's 13, we're through ...(I have a feeling that it isn't though !!!)
So...d = √[(2- 6.4)^2 + (7 + 1.8)^2] = √96.8 (< 13 !!)...that's good....and you will see why in a second...
Now....we could form two right triangles having their apexes at (2,7), their altitudes = √96.8 and with both of their hypoteneuses = 13. And we could find the length of their legs. The probem is that, we would only know two distances in either direction from (6.4, -1.8), but it wouldn't tell us what those points were. Here's a better soultion:
Imagine a circle with a radius of 13 centered at (2,7)....and we know that such a circle will have to cut the line x - 2y = 10 in two places..(because the distance from this point to the line < 13)..So....what we're really looking to do is to solve this system:
(x - 2)^2 + (y - 7) = 169
x - 2y = 10
So, using substitution, we have x = 2y + 10....and putting this into the equation of the circle, we have
((2y + 10) -2)^2 + (y-7)^2 = 169
(2y+ 8 )^2 + (y-7)^2 = 169 expand this
4y^2 + 32y + 64 + y^2 - 14y + 49 = 169 .... simplify...
5y^2 + 18y + 113 = 169
5y^2 + 18y - 56 = 0 factor this
(5y + 28) (y - 2) = 0
And setting each factor to 0, we have.......y = -28/5 = -5.6 and y = 2
So x = 2(-5.6) + 10 = -1.2 and x = 2(2) + 10 = 14
So the two points on the line that are 13 units from (2,7) are (-1.2, -5.6) and (14, 2)
Check for yourself that, in fact, both points are 13 units from (2,7) !!!
Here's a graph of the solution.........https://www.desmos.com/calculator/8gqseydskn
See Chris's answer below.
An alternative method is to notice that the points are at the intersection of the straight line y = (1/2)x - 5 and a circle of radius 13 centred on (2, 7).
Circle equation: (y - 7)2 + (x - 2)2 = 132
Replace y by (1/2)x - 5
(x/2- 5 - 7)2 + (x - 2)2 = 132
Expand the brackets and simplify
5x2/4 - 16x + 148 = 169
Subtract 169 from both sides and multiply through the result by 4
5x2 - 64x - 84 = 0
This factorises as
(5x + 6)(x - 14) = 0
so x = -6/5 (= -1.2) and x = 14
y = -(1/2)(6/5) - 5 = -28/5 = -5.6 and y = (1/2)14 - 5 = 7 - 5 = 2
So the points are (-1.2, -5.6) and (14, 2)
.
This may not be the most direct way......(someone else may nave a better idea)...well, anyway...
We have x - 2y = 10 or 2y = x - 10 or y = (1/2)x - 5
And a line perpendicular to the given one has a slope of -2
So....using (2, 7) and m = -2, we have
y = 7 = -2(x - 2)
y - 7 = -2x + 4
y = -2x + 11
And setting -2x + 11 = (1/2)x - 5 we have
-4x + 22 = x - 10 solve for x
5x = 32 → x = 32/5 = 6.4 and this is the x coordinate of the intersection of both lines
So ...the y intersection point of these two lines is -2(32/5) + 11 = -64/5 + 11 = -1.8
Now....we need to calculate the distance between this point and (2, 7)....it it's 13, we're through ...(I have a feeling that it isn't though !!!)
So...d = √[(2- 6.4)^2 + (7 + 1.8)^2] = √96.8 (< 13 !!)...that's good....and you will see why in a second...
Now....we could form two right triangles having their apexes at (2,7), their altitudes = √96.8 and with both of their hypoteneuses = 13. And we could find the length of their legs. The probem is that, we would only know two distances in either direction from (6.4, -1.8), but it wouldn't tell us what those points were. Here's a better soultion:
Imagine a circle with a radius of 13 centered at (2,7)....and we know that such a circle will have to cut the line x - 2y = 10 in two places..(because the distance from this point to the line < 13)..So....what we're really looking to do is to solve this system:
(x - 2)^2 + (y - 7) = 169
x - 2y = 10
So, using substitution, we have x = 2y + 10....and putting this into the equation of the circle, we have
((2y + 10) -2)^2 + (y-7)^2 = 169
(2y+ 8 )^2 + (y-7)^2 = 169 expand this
4y^2 + 32y + 64 + y^2 - 14y + 49 = 169 .... simplify...
5y^2 + 18y + 113 = 169
5y^2 + 18y - 56 = 0 factor this
(5y + 28) (y - 2) = 0
And setting each factor to 0, we have.......y = -28/5 = -5.6 and y = 2
So x = 2(-5.6) + 10 = -1.2 and x = 2(2) + 10 = 14
So the two points on the line that are 13 units from (2,7) are (-1.2, -5.6) and (14, 2)
Check for yourself that, in fact, both points are 13 units from (2,7) !!!
Here's a graph of the solution.........https://www.desmos.com/calculator/8gqseydskn