In the xy-coordinate plane, line L passes through the points (b, a) an

two point line equation (a,b) and (c,d)

m = slope = d-b/c-a
equation will be y-b = (d-b/c-a)(x-a)
try to form the equation of one line from a,b and c,0 and other line from c,0 an 0.,0
you will get the above equations

The standard equation of a line in x-y plane passing through points (a,b) and (c,d) :

\(\frac{y-b}{x-a} = \frac{d-b}{c-a}\).

Equation of line Y1 is created from the above equation of a line but now passing through the origin (0,0), substitute either (a,b) or (c,d) as (0,0) and you get (assuming (c,d) = (0,0))

\(\frac{y-b}{x-a} = \frac{d-b}{c-a}\) ---> \(\frac{y-b}{x-a} = \frac{0-b}{0-a}\) ---> \(\frac{y-b}{x-a} = \frac{b}{a}\)

---> After rearranging the terms, you get, \(y = x* (b/a)\)

For Y2, the OP has used the standard equation of a line (mentioned above).

Hope this helps.

Can anyone please explain how to derive b2−b∗c=b2∗(3−5√)/2?

hi zehnek , Bunuel , i understand all the steps of above solution , but if stat (1) contradicts the condition given in question of all a,b,c being different, how does that imply lines are not parallel ?
thanks in advance

Lines L and M will intersect if they are not parallel.
Parallel lines have EQUAL SLOPES.
Thus, the answer to the question stem will be NO if Lines L and M have equal slopes.
Question stem, rephrased:
Do Lines L and M have equal slopes?

Statement 1:
Let \(b=2\), with the result that \(a=\sqrt{5}-1\)
Since Line L passes through \((2, \sqrt{5}-1)\) and \((c, 0)\), the slope of Line L = \(\frac{\sqrt{5}-1-0}{2-c} = \frac{\sqrt{5}-1}{2-c}\)

Since Line M passes through \((\sqrt{5}-1, 2)\) and \((0, 0)\), the slope of Line M = \(\frac{2-0}{\sqrt{5}-1-0} = \frac{2}{\sqrt{5}-1}\)

If the two slopes are equal, we get:
\(\frac{\sqrt{5}-1}{2-c}=\frac{2}{\sqrt{5}-1}\)
\((\sqrt{5}-1)^2 = 2(2-c)\)
\(5+1-2\sqrt{5}=4-2c\)
\(2c=2\sqrt{5}-2\)
\(c=\sqrt{5}-1=a\)
Not viable, since the prompt requires that a and c be DIFFERENT nonzero numbers.

Implication:
Since it is not viable for L and M to have equal slopes, they must have DIFFERENT slopes.
Thus, the answer to the rephrased question stem is NO.
SUFFICIENT.

Statement 2:
No information about a or b.
INSUFFICIENT.

for (1) a=[(√5-1)/2]^b this is a mistake, it should write a=[(√5-1)/2]*b, or else the calculation to this question
will be more complicated since it will have to do with logarithm or exponential

(1)a=[(√5-1)/2]b


as we know that if two lines intersect, they must not parallel each other
and if lineL & lineM parallel, according to formula of slope “b/a must equal [a/(b-c)]”
thus here we only have to examine “b/a ≠ [a/(b-c)]”(true or not) in order for us to prove lines L and M intersect

if we want to examine whether under (1) the above equation true or not
first we assume b/a = [a/(b-c)]
transpose on both side --- b(b-c)=a^2
arrange the equation --- b^2=a^2+bc
substitute {[(√5-1)/2]b} for a
a^2 = {[(√5-1)/2]b}^2 = (b^2)*(1- c/b)
eliminate b^2 on both side
[(√5-1)/2] ^2 = 1- c/b
(5-2√5+1)/4 = 1- c/b
c/b = 1 - (6-2√5)/4 = 1 - (3-√5)/2 =(√5-1)/2
c =[(√5-1)/2]*b
also as we already know a=[(√5-1)/2]*b
a=c
….this totally contradict what the stimulus says “a, b, and c are different nonzero numbers”

thus if a=[(√5-1)/2]*b
b/a must not equal≠ [a/(b-c)]
lines L and M not parallel, so they must intersect
sufficient



(2) c < 0

I draw coordinate in attachment to elaborate as for why (2) is insufficient

I agree, I mistook b to be the exponent

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