# Find P and k if the equation px^(2)-8xy+3y^(2)+14x+2y+k=0 represents a pair of perpendicular lines.

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Find P and k if the equation
px^(2)-8xy+3y^(2)+14x+2y+k=0
represents a pair of perpendicular lines.

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The given equations is px^(2) - 8xy + 3y^(2) + 14x + 2y + q = 0
It represents a pair of lines perpendicular to each other .
therefore " "("coeff.of" x^(2)) +("coeff.of" y^(2)) = 0
therefore" "p+3 = 0
therefore" "p= - 3
Putting the value of p is given equaiton,
-3x^(2) - 8xy + 3y^(2) + 14x + 2y + q =0
Comparing this equation with .
ax^(2) + 2hxy + by^(2) + 2gx + 2fy + c =0 , we have
a = -3, h = - 4, b = 3, g = 7 , f =1 and c = q
therefore " "D = |{:(a,h,g),(h,b,f),(g,f,c):}|
= |{:(,-3,-4,7),(,-4,3,7),(,7,1,q):}|
 = -3 (3q -1) + 4(-4q - 7) +7(-4-21)
 = - 9q + 3 - 16q - 28 - 175
 = - 25q - 200
 = - 25(q + 8)
Since the equation represents a pair of lines,
therefore" " D = 0
rArr " " - 25 (q+8) = 0
 rArr" " q = - 8
rArr" " q = - 8
Hence, p = - 3 and q = - 8