[" A necessary and sufficient "],[" condition for the quadratic "],[" equations "a_(1)x^(2)+b_(1)x+c_(1)=0" and "],[a_(2)x^(2)+b_(2)x+c_(2)=0" to have a "],[" common root is "]

Theorem : A necessary and sufficient condition for the quadratic equations a_(1)x^(2)+b_(1)x+c_(1)=0" and "a_(2)x^(2)+b_(2)x+c_(2)=0 to have a common roots is (c_(1)a_(2)-c_(2)a_(1))^(2)=(a_(1)b_(2)-a_(2)b_(1))(b_(1)c_(2)-b_(2)c_(1))

If a_(1)x^(2)+b_(1)x+c_(1)=0 and a_(2)x^(2)+b_(2)x+c_(2)=0 has a common root,then the common root is

The line a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)=0 are perpendicular if:

Find the common factors of the expressions a_(1)x^(2)+b_(1)x+c_(1)anda_(2)x^(2)+b_(2)x+c_(1) where c_(1)ne0 .

Find the condition so that the two equations a_1 x^2+b_1 x+c_1=0 and a_2 x^2+b_2 x+c_2=0 will have a common root.

If the simultaneous linear equations a_(1)x+b_(1)y+c_(1)=0 " and " a_(2)x+b_(2)y+c_(2)=0 have only one solution, then the required condition is -

Represent the following equations in matrix form: a_(1)x+b_(1)y+c_(1)=0 a_(2)x+b_(2)y+c_(2)=0

Find the condition for two lines a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)=0 to be (i) parallel (ii) perpendicular

Find the condition for two lines a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)=0 to be (i) parallel (ii) perpendicular