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Analytical Geometry

Scorching goIITian

 Joined: 7 Oct 2009 Post: 227
14 Oct 2009 20:15:58 IST
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The ends A,B of a strait line line segment of a constant length 'c' slide upon the fixed rectangular
Engineering Entrance , JEE Main , JEE Advanced , Mathematics , Analytical Geometry

The ends A,B of a strait line line segment of a constant length 'c' slide upon the fixed rectangular axes OX & OY respectively. If the rectangle OAPB be completed then show that the locus of the foot of the perpendicular drawn from P to AB is X^2/3+y^2/3=c^2/3.

Scorching goIITian

Joined: 7 Oct 2009
Posts: 227
14 Oct 2009 20:33:42 IST
7 people liked this

Actually this question was put to me on my nudgebook

thats why i am answering here bcoz here it is presentable and others can also be benefitted.

eqn of AB is x/a + y/b = 1     ---->  bx + ay = ab

let coordinates of foot of perpendicular be (h,k)

sinc (h,k) lies on the line it should satisfy it     bh + ak = ab              --------(1)

(slope of AB) * (slope of perpendicular) = -1

so this gives a/b = (k - b)/(h - a)    =            ah - bk = a2 - b2            -----------(2)

& its given that line is of constant length c i.e.     a2 + b2 = c2       -----------(3)

solve (1) & (2) to get the value of h & k

[ This can be done by multiplying (1) by a & (2) by b  and add to get the value of h .similarly k can be found out]

h = a3 / (a2 + b2)      &           k = b3 / (a2 + b2)

using (3) we get h =  a3 /c2          and   k = b3/c2

a = (hc2)1/3     b = (kc2)1/3

put in (3)

(hc2)2/3 + (kc2)2/3 = c2

h2/3 + k2/3 = c2/3

put h = x and k = y

x2/3 + y2/3 = c2/3

## Hence proved

Blazing goIITian

Joined: 15 Aug 2008
Posts: 584
31 Oct 2009 14:25:37 IST
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The ends A,B of a strait line line segment of a constant length 'c' slide upon the fixed r

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