Circle 4 Question 21

21. Let a given line L1 intersect the X and Y-axes at P and Q respectively. Let another line L2, perpendicular to L1, cut the X and Y-axes at R and S, respectively. Show that the locus of the point of intersection of the line PS and QR is a circle passing through the origin.

(1987, 3M)

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Solution:

  1. Let the equation of L1 be xcosα+ysinα=p1.

Then, any line perpendicular to L1 is

xsinαycosα=p2

where, p2 is a variable.

Then, L1 meets X-axis at P(p1secα,0) and Y-axis at Q(0,p1cosecα).

Similarly, L2 meets X-axis at R(p2cosecα,0) and Y-axis at S(0,p2secα).

Now, equation of PS is,

xp1secα+yp2secα=1xp1yp2=secα

Similarly, equation of QR is

xp2cosecα+yp1cosecα=1xp2+yp1=cosecα

Locus of point of intersection of PS and QR can be obtained by eliminating the variable p2 from Eqs. (i) and (ii).

xp1secαxy+yp1=cosecα(xp1secα)x+y2=p1ycosecαx2+y2p1xsecαp1ycosecα=0 which is a circle through origin. 



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