Intersection of curve and line

Question

This is a question which I want to solve, taken from this sample question paper for an exam I'm appearing for tommorow:

If a line, parallel to, but not identical with, x- axis cuts the graph of the curve $$y={(x-1)}/({(x-2)(x-3)})$$ at $x=a$ and $x=b$ then evaluate $$(a-b)(b-1)$$

Here is what I did:
Let the line be $y=c$ (since it is parallel to x-axis) . Then $c=(a-1)/((a-2)(a-3))$ and similarly for b .
But then what ? Please help.

score 1 · Answer 1 · answered Nov 16 '13 at 12:54

1

Assuming $(x-2)(x-3)\ne0$

So, $$c=\frac{x-1}{(x-2)(x-3)}\implies cx^2-(5c+1)x+6c+1=0$$ which is a Quadratic Equation in $x$ whose roots are $a,b$

If $c=0,x=1$ for finite $x\implies a=b$

Else $$ a+b=\frac{5c+1}c=5+\frac1c\text{ and }ab=\frac{6c+1}c=6+\frac1c$$

I think proposition in the question should be the one available by eliminating the foreign(which we have introduced in the question) element $c$

answered Nov 16 '13 at 12:54

lab bhattacharjee

274,582

@AGoogler,the elimination gives $$ab-(a+b)=1\implies (a-1)(b-1)=2$$ can you please verify the question? – lab bhattacharjee Nov 16 '13 at 14:24
I verified the question . Since this is a subjective question , maybe I'm allowed to write the answer in terms of c. – A Googler Nov 16 '13 at 14:37

Intersection of curve and line

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