If `sin(x+y)=p/(sqrt(1+p^2)) and cos(x-y)=1/(sqrt(1+q^2))` then show that `tan x` is a root of the equation `(p+q)z^2+2(1-pq)z-(p+q)=0`

If Sin^(-1)(2p)/(1+p^(2)) - Cos^(-1)((1-q^(2))/(1+q^(2))) = Tan^(-1) (2x)/(1-x^(2)) , then prove that x = (p-q)/(1+pq)

If the roots of the equation p(q-r)x^2+q(r-p)x+r(p-q)=0 be equal ,show that 1/p+1/r=2/q

If x^2+p x+q=0a n dx^2+q x+p=0,(p!=q) have a common roots, show that 1+p+q=0 . Also, show that their other roots are the roots of the equation x^2+x+p q=0.

If x^2+p x+q=0a n dx^2+q x+p=0,(p!=q) have a common roots, show that 1+p+q=0 . Also, show that their other roots are the roots of the equation x^2+x+p q=0.

If the roots of the equation p(q-r)x^2+q(r-p)x+r(p-q)=0 be equal, show that 1/p+1/r=2/q .

If the equation x^(2)+px+q=0 and x^(2)+p'x+q'=0 have a common root show that it must be equal to (pq'-p'q)/(q-q') or (q-q')/(p'-p)

If p+2sqrt(x-1)=q , and q > p , what is x-1 in terms of p and q ?

If (x_1, y_1)&(x_2, y_2) are the ends of diameter of a circle such that x_1&x_2 are the roots of the equation a x^2+b x+c=0 and y_1&y_2 are the roots of the equation p y^2+q y+c=0 . Then the coordinates f the centre of the circle is: (b/(2a), q/(2p)) (b) (-b/(2a),=q/(2p)) (b/a , q/p) (d) None of these

If p and q are the roots of the equations x^(2) - 3x+ 2 = 0 , find the value of (1)/(p) - (1)/(q)