If `a ,b ,p ,q` are non zero real numbers, then how many comman roots would two equations: `2a^2x^2-2a b x+b^2=0 and p^2x^2+2p q x+q^2=0` have?

If p , q , ra n ds are real numbers such that p r=2(q+s), then show that at least one of the equations x^2+p x+q=0 and x^2+r x+s=0 has real roots.

If p ,q are real p!=q , then show that the roots of the equation (p-q)x^2+5(p+q)x-2(p-q)=0 are real and unequal.

If p , qr are real and p!=q , then show that the roots of the equation (p-q)x^2+5(p+q)x-2(p-q=0 are real and unequal.

Column I, Column II If a ,b ,c ,a n dd are four zero real numbers such that (d+a-b)^2+(d+b-c)^2=0 and he root of the equation a(b-c)x^2+b(c-a)x=c(a-b)=0 are real and equal, then, p. a+b+c=0 If the roots the equation (a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0 are real and equal, then, q. a ,b ,ca r einAdotPdot If the equation a x^2+b x+c=0a n dx^3-3x^2+3x-0 have a common real root, then, r. a ,b ,ca r einGdotPdot Let a ,b ,c be positive real number such that the expression b x^2+(sqrt((a+b)^2+b^2))x+(a+c) is non-negative AAx in R , then, s. a ,b ,ca r einHdotPdot

If the abscissa and ordinates of two points P and Q are the roots of the equations x^2+2a x-b^2=0 and x^2+2p x-q^2=0 , respectively, then find the equation of the circle with P Q as diameter.

If the abscissa and ordinates of two points Pa n dQ are the roots of the equations x^2+2a x-b^2=0 and x^2+2p x-q^2=0 , respectively, then find the equation of the circle with P Q as diameter.

If the abscissa and ordinates of two points Pa n dQ are the roots of the equations x^2+2a x-b^2=0 and x^2+2p x-q^2=0 , respectively, then find the equation of the circle with P Q as diameter.

If p,q,r,s are real and prgt4(q+s) then show that at least one of the equations x^2+px+q=0 and x^2+rx+s=0 has real roots.

Show that if p,q,r and s are real numbers and pr=2(q+s) , then atleast one of the equations x^(2)+px+q=0 and x^(2)+rx+s=0 has real roots.

If the normals to the parabola y^2=4a x at three points (a p^2,2a p), and (a q^2,2a q) are concurrent, then the common root of equations P x^2+q x+r=0 and a(b-c)x^2+b(c-a)x+c(a-b)=0 is p (b) q (c) r (d) 1