Find the condition if the roots of `a x^2+2b x+c=0a n db x^2-2sqrt(a c )x+b=0` are simultaneously real.

Find the condition if the circle whose equations are x^2+y^2+c^2=2a x and x^2+y^2+c^2-2b y=0 touch one another externally.

If s intheta,costheta be the roots of a x^2+b x+c=0 , then prove that b^2=a^2+2ac.

The common root of the equation x^(2) - bx + c = 0 and x^(2) + b x - a= 0 is ………..

Find the condition on a , b ,c ,d such that equations 2a x^3+bx^2+c x+d=0a n d 2ax^2+3b x+4c=0 have a common root.

x_1a n dx_2 are the roots of a x^2+b x+c=0a n dx_1x_2<0. Roots of x_1(x-x_2)^2+x_2(x-x_1)^2()_()=0 are a. real and of opposite sign b. negative c. positive d. none real

If the roots of (a-b)x^(2)+(b-c)x+(c-a)=0 are equal, prove that 2a=b+c .

If aa n dc are the lengths of segments of any focal chord of the parabola y^2=2b x ,(b >0), then the roots of the equation a x^2+b x+c=0 are (a)real and distinct (b) real and equal (c)imaginary (d) none of these

If tanthetaa n dsectheta are the roots of a x^2+b x+c=0, then prove that a^4=b^2(b^2-4ac)dot

If a, b are real then show that the roots of the equation (a-b)x^(2)-6(a+b)x-9(a-b)=0 are real and unequal.

Let a circle be given by 2x(x-1)+y(2y-b)=0,(a!=0,b!=0) . Find the condition on aa n db if two chords each bisected by the x-axis, can be drawn to the circle from (a , b/2)