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let agiven line L1 intersect the x and y axis at P &Q respectively.Let another line L2 perpendicular L1 cut the x and y axis at R & S respectively.Show that the locus of the point of intersection of the lines PS & QR is a circle passing through the origin.
let given line is L1: y=mx+c
(P,Q)=(-c/m,0) , (0,c)
now L2 is perpendicular to line L1
slope of L2*slope of L1=-1 (m1m2=-1)
eq of L2: y=-x/m + k (k is any variable)
(Q,R)=(km,0),(0,k)
eq of PS is y=mkx/c + k and.................1
eq of QR is y=-x/km + c ...................2
let the point if intersection be (X,Y)
after solving 1 and 2
X= (c-k)/(mk/c + 1/mk )...............3
Y= (km+1/m) /(mk/c + 1/mk)...........4
eliminate k by solving eq 3 and 4 you will get the equation in terms of X,Y and c .......this will give the required locus which is circle passing through origin...