If the equations `ax + by = 1` and `cx^2 +dy^2=1` have only one solution, prove that `a^2/c+ b^2/d = 1` and `x=a/c`,`y=b/d`

The equations x^2+b^2=1-2bx and x^2+a^2=1-2ax have only oneroot in common then |a-b|=

If the equation ax^2+2bx+c=0 and ax^2+2cx+b=0 b!=c have a common root ,then a/(b+c =

If a,b,c, a_1,b_1,c_1 are rational and equations ax^2+2bx+c=0 and a_1x^2+2b_1x+c_1=0 have one and only one root in common, prove that b^2-ac and b_1^2-a_1c_1 must be perfect squares.

If the two equations x^2 - cx + d = 0 and x^2- ax + b = 0 have one common root and the second equation has equal roots, then 2 (b + d) =

Number of triplets of a, b & c for which the system of equations, ax - by = 2a - b and (c +1)x+cy = 10-a + 3 b has infinitely many solutions and x = 1. y = 3 is one of the solutions, is: (A) exactly one (B) exactly two (C) exactly three (D) infinitely many

If the quadratic equation ax^(2) + 2cx + b = 0 and ax^(2) + 2x + c = 0 ( b != c ) have a common root then a + 4b + 4c is equal to

If one of the roots of the equation ax^2 + bx + c = 0 be reciprocal of one of the a_1 x^2 + b_1 x + c_1 = 0 , then prove that (a a_1-c c_1)^2 =(bc_1-ab_1) (b_1c-a_1 b) .

If 3(a+2c)=4(b+3d), then the equation a x^3+b x^2+c x+d=0 will have (a) no real solution (b) at least one real root in (-1,0) (c) at least one real root in (0,1) (d) none of these

If 1,2,3 and 4 are the roots of the equation x^4 + ax^3 + bx^2 +cx +d=0 then a+ 2b +c=

If a,b,c and d are in G.P , then show that ax^(2)+c divides ax^(3)+bx^(2)+cx+d