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# A Wonderful Question

if a,b,c are distinct natural numbers greater than 1 and a+b+c+ab+bc+ca+abc=1000 where c>b>a.If three straight lines are L1:bx+ay-ab=0, L2:cx+by-bc=0 and L3:ax+cy-ac=0 then sum of intercepts of line L1, L2 and L3 on the axes are ?

Note by Iq 131
4 years, 7 months ago

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The only hard part of this question is to figure out what $$a, b, c$$ are, which is a problem proposed by Vishnoo P. several weeks back.

You may view the problem by clicking on this link. If you received the problem then, you will be directed to the solution presented by Rahul N..

Note that this question by Manish is also badly phrased, since it's unclear what "sum of intercepts of lines ... on the axes" refers to. Do you want both the x and the y axis?

In fact, a much better question would be to ask: The line $$\ell: bx+ay-ab=0$$ intersects the x-axis at $$(x,0)$$ and the y-axis at $$(0,y)$$. What is the value of $$x+y$$?

Staff - 4 years, 7 months ago

sum of axes is =2(a+b+c)=56

- 4 years, 7 months ago

=>a+b+c+ab+bc+ca+abc=1000 =>(a+1)(b+1)(c+1)=1001=(7)(11)(13) =>Hence a=6,b=10,c=12

- 4 years, 7 months ago

It means both the axes actually.

- 4 years, 7 months ago

What do you mean, the sum on the axes?

- 4 years, 7 months ago