# Find ' ' P and ' ' Q If the Equation - Mathematics and Statistics

Sum

Find 'p' and 'q' if the equation px2 - 8xy +3y2 +14 x +2y +q = 0 represents a pair of perpendicular lines.

#### Solution

Given general equation is px2 - 8xy +3y2 +14 x +2y +q = 0

a = p , h = -4 , b = 3 , g = 7, f = 1 , c = q

Since the equation represents perpendicular lines

a +b =0

p + 3 = 0

∴ p = -3

∴ a = p = -3

Since the equation represents a pair of lines.

abs[("a","h" ,"g"),("h","b","f"),("g" ,"f" , "c")] = 0

abs[(-3,-4,7),(-4,3,1),(7,1,"q")] = 0

∴ -3(3q-1)+4(-4q-7)+7(-4-21)=0

∴ -9q +3 -16q - 28 - 28 - 147 = 0

∴ -25q + 3 - 28 - 175 = 0

∴ -25q + 3 - 203 = 0

∴ -25q - 200 = 0

∴ -25q = 200

∴ q = 200/-25

∴ q = -8

∴ p = -3 , q = -8

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
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