A transvers axis cuts the same branch of a hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` at `Pa n dP '` and the asymptotes at `Q` and `Q '` . Prove that `P Q=P ' Q '` and `P Q^(prime)=P^(prime)Qdot`

A transversal cut the same branch of a hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 in P,P' and the anymptotes in Q.Q' ,the value of (PQ+PQ')-(P'Q'-P'Q)

A straight line intersects the same branch of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 in P_(1) and P_(2) and meets its asymptotes in Q_(1) and Q_(2) . Then, P_(1)Q_(2)-P_(2)Q_(1) is equal to

If any line perpendicular to the transverse axis cuts the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 and the conjugate hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=-1 at points P and Q, respectively,then prove that normal at P and Q meet on the x-axis.

If P + Q = R and P - Q = S, prove that R^2 + S^2 = 2(P^2 + Q^2)

If the roots of the equation (1-q+(p^(2))/(2))x^(2)+p(1+q)x+q(q-1)+(p^(2))/(2)=0 are equal then show that p^(2)=4q

Consider the family of all circles whose centers lie on the straight line y=x . If this family of circles is represented by the differential equation P y^(primeprime)+Q y^(prime)+1=0, where P ,Q are functions of x , y and y^(prime)(h e r ey^(prime)=(dy)/(dx),y^=(d^2y)/(dx^2)), then which of the following statements is (are) true? (a)P=y+x (b)P=y-x (c)P+Q=1-x+y+y^(prime)+(y^(prime))^2 (d)P-Q=x+y-y^(prime)-(y^(prime))^2

If p, q, r are prime numbers such that p^(2)-q^(2)=r then which of the following options are correct ?

If p and q are the intercept on the axis cut by the tangent of sqrt(((x)/(a)))+sqrt(((y)/(b)))=1, prove that (p)/(a)+(q)/(b)=1

If p and q are the intercept on the axis cut by the tangent of sqrt(((x)/(a)))+sqrt(((y)/(b)))=1, prove that (p)/(a)+(q)/(b)=1