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#1 2005-11-21 06:45:58
- Sephiroth Valentine
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Stuck on a difficult co-ordinate geomatry problem
6(C) A line containg the point(-2,4) has a slope of m which is not equal to zero. The line intercepts the x-axis at (x1,0) and the y axis at (0,y1) If x1 + y1=-4m find the slopes of the two lines that satisfy this condition. Hence find the tangent of the acute angle between these two lines.
I really don't know where to start Tried using the slope formula y2-y1/x2-x1 using the point (-2,4) and (x1,0) and then I used the points (-2,4) and (0,y1) to get another value for m. I put bothe equal to one another but I could'nt get a value to work out. Could any help me please.
#2 2005-11-21 08:14:33
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Re: Stuck on a difficult co-ordinate geomatry problem
I think it is just a bunch of formulas to solve.
Using the Line Equation: y = mx+b
At (-2,4): 4 = m(-2) + b → b = 4+2m
At (x1,0): 0 = m(x1) + b → x1 = -b/m
At (0,y1) y1 = m(0) + b → y1 = b
And we are also given: x1+y1 = -4m
So we have four equations, and four unknowns (m,b,x1,y1), so now we can hopefully just use algebra to solve it.
There may be a quicker way to do this, but I am just going to start anywhere and see what happens:
Start with: x1+y1 = -4m
Substitute for x1 and y1: -b/m + b = -4m
Looks "quadratic", so multiply by m: -b + bm = -4m²
Put in quadratic form: 4m² +bm - b = 0
Replace b with (4+2m): 4m² +(4+2m)m - (4+2m) = 0
Expand: 4m² + 4m + 2m² - 4 - 2m = 0
Collect terms: 6m² + 2m - 4 = 0
Now just solve the quadratic equation here and you get:
m = 2/3 or -1
I will leave you to check that this actually works (my error rate is about 1 in 10 
)
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#3 2005-11-21 08:45:08
- Sephiroth Valentine
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Re: Stuck on a difficult co-ordinate geomatry problem
Something seems worng as I checked the answer at the back of the book and the answers for m were supposed to be either -1 or 2
#4 2005-11-21 08:48:05
- Sephiroth Valentine
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Re: Stuck on a difficult co-ordinate geomatry problem
Sorry I see what was wrong I made a mistake when typing the question. I should have been x1 + y1=-4 not x1 + y1=-4m . Thanks for all your help though. Can your method still work with this equation?
#5 2005-11-21 16:59:32
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Re: Stuck on a difficult co-ordinate geomatry problem
Have a go ... copy and paste what I did and make the needed changes - I think you will still end up with a quadratic equation. Post your answer here so I can see.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#6 2005-11-22 04:19:45
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Re: Stuck on a difficult co-ordinate geomatry problem
At (-2,4): 4 = m(-2) + c → c = 4+2m
At (x1,0): 0 = m(x1) + c→ x1 = -c/m
At (0,y1) y1 = m(0) + c → y1 = c
x1+y1 = -4
∴-c/m +c=-4/1
y1=c
c=2m+4
next substitute c value for c into equation: -c/m +c=-4/1
(-2m+4)/m +2m+4=-4
2m+4+2m²+4m=0
since 2m+4=c==y1
then x1+4= 2m²+4m
I still can't get it out any ideas anyone. I guess I went hopelessly wrong
#7 2005-11-22 08:58:01
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Re: Stuck on a difficult co-ordinate geomatry problem
At (-2,4): 4 = m(-2) + b → b = 4+2m
At (x1,0): 0 = m(x1) + b → x1 = -b/m
At (0,y1) y1 = m(0) + b → y1 = b
And we are also given: x1+y1 = -4
So we have four equations, and four unknowns (m,b,x1,y1), so now we can hopefully just use algebra to solve it.
Start with: x1+y1 = -4
Substitute for x1 and y1: -b/m + b = -4
Multiply by m: -b + bm = -4m
Put at one side: 4m + bm - b = 0
Replace b with (4+2m): 4m +(4+2m)m - (4+2m) = 0
Expand: 4m + 4m + 2m² - 4 - 2m = 0
Collect terms: 2m² + 6m - 4 = 0
Now just solve the quadratic equation here and you get:
m = 0.56 and -3.56
Actual values are (-6±√68)/4
NOW, let us check the answer (just the 0.56 one):
b = 4+2m = 5.12
At x=-2, y=0.56(-2)+5.12 = 4
At x=0, y1 = 0.56(0)+5.12 = 5.12
At y=0, 0 = 0.56(x1)+5.12, hence x1 = -5.12/0.56 = -9.14
x1+y1 = 5.12-9.14 = -4.02 (close enough to -4), so that looks right!
If this doesn't agree with the answer you have, then ... who knows!
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman