If `(x - alpha)` be a factor common to `a_(1)x^(2) + b_(1)x +c` and `a_2x^2 + b_(2)x + c`, then prove that: `alpha(a_(1)-a_(2))=b_(2)-b_(1)`

If two equation a_(1) x^(2) + b_(1) x + c_(1) = 0 and, a_(2) x^(2) + b_(2) x + c_(2) = 0 have a common root, then the value of (a_(1) b_(2) - a_(2) b_(1)) (b_(1) c_(2) - b_(2) c_(1)) , is

If the points (a_(1),b_(1))m(a_(2),b_(2))" and " (a_(1)-a_(2),b_(2)-b_(2)) are collinear, then prove that a_(1)/a_(2)=b_(1)/b_(2)

If the roots of a_(1)x^(2)+b_(1)x+c_(1)=0 are alpha_(1),beta_(1) and those of a_(2)x^(2)+b_(2)x+c_(2)=0 are alpha_(2),beta_(2) such that alpha_(1)alpha_(2)=beta_(1)beta_(2)=1 then

If (a_(1))/(a_(2)),(b_(1))/(b_(2)),(c_(1))/(c_(2)) are in AP and the equations a_(1)x^(2)+2b_(1)x+c_(1)=0 and a_(2)x^(2)+2b_(2)x+c_(2)=0 have a common root, prove that a_(2),b_(2),c_(2) are in G.P

For two linear equations a_(1)x + b_(1)y + c_(1)= 0 and a_(2) x+ b_(2)y+ c_(2)= 0 , then condition (a_(1))/(a_(2)) = (b_(1))/(b_(2))= (c_(1))/(c_(2)) is for

Suppose two quadratic equations a_(1)x^(2)+b_(1)x+c_(1)=0 and a_(2)x^(2)+b_(2)x+c_(2)=0 have a common root alpha , then a_(1)alpha^(2)+b_(1)alpha+c_(1)=0 " " …..(1) and a_(2)alpha^(2)+b_(2)alpha+c_(2)=0 " " ……(2) Eliminating alpha using crose - multiplication method gives us the condition for a common root Solving two equations simultaneously, the common root can be obtained. Now, consider three quadratic equations, x^(2)-2r p_(r )x+r=0, r=1, 2, 3 given that each pair has exactly one root common. In the notations of above problem, (gamma)/(alpha) is equal to

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 .