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Q.

Show that the product of the perpendicular distances from the origin to the pair of straight lines represented by ax2+2hxy+by2+2gx+2fy+c=0 is |c|(ab)2+4h2 

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Detailed Solution

Given equation
S=ax2+2hxy+by2+2gx+2fy+c=0
Let the lines represented by S = 0 be
l1x+m1y+n1=0 ...(1)
and  l2x+m2y+n2=0 (2)
ax2+2hxy+by2+2gx+2fy+c=l1x+m1y+n1l2x+m2y+n2
Comparing the coefficients of like terms we get
l1l2=a;m1m2=b;n1n2=cl1m2+m1l2=2h,l1n2+n1l2=2g,m1n2+m2n1=2f
Perpendicular distance from origin to the line  (1) is n1l12+m12
Perpendicular distance from origin to the line (2) is n2l22+m22
Product of the perpendicular distance from (0, 0) to S = 0 is
n1l12+m12n2l22+m22=n1n2l12l22+m12l22+l12m22+m12m22=|c|l1l2m1m22+2l1l2m1m2+l1m2+m1l22-2l1l2m1m2=|c|(ab)2+(2h)2=|c|(ab)2+4h2
Hence proved

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