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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.


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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)24.2(3y2+7y+4)4

    =(5y+6)±25y2+60y+3624y256y324

      X=(5y+6)±y2+4y+44

X=(5y+6)±(y+2)4

X=5y6+y+24,5y6y24 

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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