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Question

Find p and q if equation px28xy+3y2+14x+2y+q=5, represents pair of perpendicular lines.

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Solution

Given, px28xy+3y2+14x+2y+q=5...(i) represents pair of
perpendicular lines.
equation (i) is of the form
ax2+2hxy+by2+2gx+2fy+c=0
a=p,h=4,b=3,g=7,f=1,c=q5.
since the equation (i) represents pair of perpendicular lines
a+b=0
p+3=0
p=3....(ii)
also eqn (i) represents a pair of straight lines
∣ ∣ahghbfgfc∣ ∣=0
∣ ∣p4743171q5∣ ∣=0
p(3q151)+4(4q+207)+7(421)=0
(3)(3q16)+4(4q+13)175=0 [from (ii)P=-3]
9q+4816q+52175=0
25q75=0
25q=75
q=3
Hence p=3,q=3.

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