7: IIT level solved IV

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We are dealing with the concept of SERIES here. It is quite simple actually. Instead of writing many equations in sequence, we write one i.e. x=r where r takes many values i.e. r= 1,2,3…. You write x=r simply but it means many straight lines. 
As given each of these lines parallel to the y axis cuts a line, L passing through the Origin namely y=(sqrt 5)*x and thus the points where these numerous lines cuts this line L has been tabulated. The answer is in the form of a series too as we have shown int the diagram. Thereafter to find the sum of squares of the distances from Origin to these points we end up with a Sum of a Series and we use a simple formula to get the answer. 


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This is a very interesting question. Two random lines are taken, they cut the axes in 4 points as usual. 
The additional fact is that these four points are concyclic. That is what we have to use. 

So, I drew the two lines and drew the circle. We have to use the property of the circle. That is obvious. 

Suddenly it flashes as you look at the circle and the two lines. Like AB & CD, AC & BD are also two chords of the circle and we have a rule that if two chords cut each other, the product of the segments of the chords are equal. THAT gives us an equation which helps us solve the problem. 


A Problem on Locus (below)

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Any line through the Origin is y=mx and that cuts the other two parallel lines given. So by equating the lines we obtain points A and B and thus we also find OA & OB. 
As a technique to find the locus of a Point P we take tis coordinates temporarily as (h,k) then we establish the relationship that is given or sometimes, like in our previous example, from a geometrical property that emerges from the diagram given. Finally, once we use that property and do the algebra we end up with a relationship between h and k. It is then that we replace h by x and k by y and obtain the final locus. 


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Algebra is used to find unknown quantities. Look at the last sentence in the question. It says find the slope i.e. find m. So, we require an equation with m. So, we look at the information given:

a line passes through (1,2) and its slope is m. And the rest of the information we use to build up the equation containing m and thus find the same.