HARSHIT B.

asked • 12/18/16

if p(att,2at),q(a/tt, -2a/t) and s( a,0) be any three points, show that 1/sp+ 1/ sq is independent of t

It is a question of coordinate geometry
Here tt means t SQUARE

Kenneth S.

Conventionally, points are given Capital Letters, and if you don't know how to present t squared with an actual exponent, then you should type t^2.  As given, this problem is unattractively presented.
 
It appears that you should compute distances SP and SQ, using the distance formula, and then do the necessary algebra; I do not volunteer, however. Why don't youl try it, using these suggestions?
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12/18/16

HARSHIT B.

I already solved distance between so and sq. But I can't solve algebraic problem
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12/18/16

1 Expert Answer

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Mark M. answered • 12/18/16

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P = (at2, 2at)   Q = (a/t2, -2a/t)   S = (a,0)
 
SP = √[(a-at2)2 + (2at-0)2] = √[a2(1-t2)2 + 4a2t2]
 
                                       = √[a2(1-2t2+t4+4t2)]
 
                                       = lal√(t4+2t2+1)  
 
                                       = lal√(t2+1)2 = lal(t2+1)
 
SQ = √[(a-a/t2)2 + (-2a/t-0)2]
 
     = √[a2(1-1/t2)2 +4a2/t2]
 
     = √[a2(1-2/t2+1/t4+4/t2)]
 
     = lal√(1+2/t2+1/t4)  
 
     = lal√[(t4+2t2+1)/t4)] = lal√[(t2+1)2/t4]
 
                                   = lal(t2+1)/t2
 
So, 1/SP + 1/SQ = 1/(lal(t2+1)) + t2/(lal(t2+1))
 
                        = (1+t2)/(lal(1+t2)) = 1/lal ←independent of t!!
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