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Q.

Let A  and  B  be any two points on y-axis and the line  y=2x respectively. Also P  and Q   be the feet of perpendicular from A  and B  on the line y=2x  and y-axis respectively. If AB  passes through the points  (1,3), then the line PQ  passes through fixed point (x1,y1)  such that y1x1  is

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answer is 7.

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Detailed Solution

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Let A(0,a)   and  B(b,2b)
Coordinate of  P  is  (2a5,4a5)
Coordinate of Q  is  (0,2b)
Equation of AB   is ya(2ab)x=0 
     It passes through  (1,3)  hence  a=bb1
Also equation of PQ   is  y2b=(25ba)x
i.e., y7x+b(x2)=0

     Fixed point is  (25,145)

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