Show that the expression `((ax-b)(dx-c))/((bx-a)(cx-d))` will be capable of all values when x is real if `a^2-b2` and `c^2-d^2` have the same sign

Show that the expression ((ax-b)(dx-c))/((bx-a)(cx-d)) be capabie of all values when x is real, if a^2 - b^2 and c^(2)-d^(2) have the same sign.

Show that the expression ((ax-b)(dx-c))/((bx-a)(cx-d)) will be capable of all values when x is real,if (a^(2)-b^(2)) and (c^(2)-d^(2)) ahve the same sign.

Solve the equations : (b-ax)/(bx-a)=(d-cx)/(dx-c)

Theorem : If the roots of ax^(2) + bx+c=0 are real and equal to alpha =(-b)/(2a) , then for alpha ne x in R, ax^(2) + bx+c and 'a' have the same sign.

If (a-bx)/(a+bx)=(b-cx)/(b+cx)=(c-dx)/(c+dx) show that a,b,c , d are in G.P.

Theorem : Let a, b, c in R and a ne 0. Then the roots of ax^(2)+bx+c=0 are non-real complex numbers if and only if ax^(2)+bx+c and a have the same sign for all x in R .

If (a+bx)/(a-bx)=(b-cx)/(b-cx)=(c+dx)/(c-dx)( x ne 0) then show that a, b, c and d are in G.P.

If (a+bx)/(a-bx)=(b-cx)/(b-cx)=(c+dx)/(c-dx)( x ne 0) then show that a, b, c and d are in G.P.

If (a+bx)/(a-bx)=(b-cx)/(b-cx)=(c+dx)/(c-dx)( x ne 0) then show that a, b, c and d are in G.P.

Let a, b, c be three distinct real numbers such that each of the expressions ax^+bx +c, bx^2 + cx + a and ax^2 + bx + c are positive for all x in R and let alpha=(bc +ca+ab)/(a^2+b^2+c^2) then (A)alpha 1/4 (D) alpha>1