  Question

Equation 2x2+5xy+3y2+6x+7y+4=0 represents a pair of straight lines. If their point of intersection is (h,k) find the value of h-k. __

Solution

It is given that equation 2x2+5xy+3y2+6x+7y+4=0 represents the pair of straight lines We can write this equation as quadratic equation in x We have,                  2x2+(5y+6)x+3y2+7y+4=0-----------------------(1) Roots of the given equation                                           X=−(5y+6)±√(5y+6)2−4.2(3y2+7y+4)4     =−(5y+6)±√25y2+60y+36−24y2−56y−324       X=−(5y+6)±√y2+4y+44 X=−(5y+6)±(y+2)4 X=−5y−6+y+24,−5y−6−y−24  Or      4x = -4y - 4  ,          4x + 6y + 8 = 0           X + y + 1 = 0   and  2x + 3y + 4 = 0 Hence, equation 1 represents the pair of straight lines whose equations are                       X + y + 1 = 0------------------2                      2x + 3y + 4= 0-----------------3 Multiplying 2 in equation 2 and subtracting from equation 3                  2x + 2y + 2 = 0                  2x + 3y + 4 = 0                   -y - 2 = 0                    -y = 2                      y = -2            Substituting in equation 1 -2+x+1=0 X=1 Point of intersection (1,-2) h=1,       k=-2 h-k=       1-(-2)=3  Mathematics

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