Simple coordinate geomety problem

[silent_hunter]

I'm not sure where to post this. So I'm posting in the General Math section.This is a simple coordinate geometry problem: We have to find the equation of line(s) passing through the point (7,17) and having a distance of 6 units from the point(1,9).

Now I'm posting my approach:
the equation of line passing through(7,17) and having slope m is
y-17=m(x-7)
or, mx-y-7m-17=0
now the line have a distance of 6 units from point(1,9).
so \frac{m-9-7m+17}{\sqrt{m^2 +1}}=\pm6
or,(6m-8)² =36(m² +1)
or, m=7/24
so the equation of line becomes 7x-24y+359 =0
This way we get only line.But if you draw it in a graph paper,you'll see that there should be another line which is x-7=0 which is parallel to the y-axis.(draw a circle of radius 6 from (1,9) and then draw tangent from (7,17) to the circle).
I think we can't get the second one because it has undefined slope.
My question is how can I get the second line without plotting in graph?

 

[tiny-tim]

hi silent_hunter!

(try using the X² button just above the Reply box )

you lost half the solutions when you took the square-root of this line …

or,(6m-8)^2 =36(m^2 +1)
or, m=7/24

you should have put the intermediate step (6m-8) = ±36(m² +1)

 

[silent_hunter]

you should have put the intermediate step (6m-8) = ±36(m² +1)

thanks for your reply
(6m-8)² =36(m² +1)
in that step I squared both sides of the equation ,so the ± sign should go away.

 

[tiny-tim]

sorry, i got that wrong

this is where you missed a solution …

the equation of line passing through(7,17) and having slope m is
y-17=m(x-7)
or, mx-y-7m=17=0

… you missed out m = ∞ ! ​

 

[silent_hunter]

sorry, i got that wrong

this is where you missed a solution …

… you missed out m = ∞ ! ​

sorry I made a typing mistake.
Its mx-y-7m-17=0 but I didn't get what you said. Would you please elaborate?Thanks.

 

[tiny-tim]

one of the two lines is x = 7, isn't it?

(with slope ∞)

that doesn't come up for any m in mx-y-7m-17=0

 

[kdetro]

if you think, (y-9)^2+(x-1)^2=36 then

if you think of it as (y-9)^2 + (x-1)^2 = 36

then:

y=cos(3(x-1))+15.91 or so should hit twice.

I'll bet some kind of absolute value function would hit three times.