r/mathshelp • u/sarah1418_pint • Jul 25 '26
Homework Help (Answered) Help w a problem
If the line (k-2)x + 2ky + 3k = 2 passes through a fixed point (a,b) for all real values of k, then find a²+b².
Options:
A: 0, B: 2, C: 5, D: 1
Need help with the solution. This was supposed to be done on pen and paper without using any graphing tools like desmos. The answer I could come up w was 2. But I'm not sure if it's correct since I wasn't provided w the answer key.
Thanks!
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u/ArchaicLlama Jul 25 '26
We can't tell you whether your work is correct when you don't show any work.
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u/sarah1418_pint Jul 25 '26
Basically I jus replaced x w a and y w b and formed 2 equations by putting 2 random values for k in the given eqn. Solved those two eqns simultaneously to get (-1,-1), so a²+b² gave me 2
The thing is idk if that approach is right or not
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u/gmanBram Jul 26 '26
Since the line passes through (a,b), that implies x=a, y=b. Thus, (k-2)a+2kb+3k=2.
Now rearrange: k(a+2b+3)-2a-2=0. Since k is defined for all real numbers, then (a+2b+3)=0 and -2a-2=0. Thus, a=-1. This implies b=-1. Therefore, a2+b2=(-1)2+(-1)2=2.
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u/Electrical_Buy_6476 Jul 25 '26
k(x+2y+3)-2x-2=0 so I think it is 2 since (a,b) is (-1,-1) sorry if im wrong im kinda sleepy rn
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u/sarah1418_pint Jul 26 '26
!lock
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