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`

Any tangent to an ellips (x^(2))/(a^(2))+(y^(2))/(b^(2))=1, (a gt b) cut by the tangents at the end points of major axis is T and T' . Prove that the length of intercept of the major axis cut by the circle describe on T T' as diameter is constant and equal to 2sqrt(a^(2)-b^(2)) .

Two circles cut at A and B and a straight line PAQ cuts the circles at P and Q.If the tangents at P and Q intersects at R, prove that R,PB,Q are concyclic.

Let P and Q be two points on the minor axis each at distance sqrt(a^(2)-b^(2))(a>b) from the centre of the ellipse and p and q be the lengths of the perpendiculars upon any tangent from P and Q then find the value of p^(2)+q^(2)

The value of sqrt(p^(-1)q).sqt(q^(-1)r).sqrt(r^(-1) p) is

Locus of the middle points of the line segment joining P(0, sqrt(1-t^2) + t) and Q(2t, sqrt(1-t^2) - t) cuts an intercept of length a on the line x+y=1 , then a = (A) 1/sqrt(2) (B) sqrt(2) (C) 2 (D) none of these

Locus of the middle points of the line segment joining P(0, sqrt(1-t^2) + t) and Q(2t, sqrt(1-t^2) - t) cuts an intercept of length a on the line x+y=1 , then a = (A) 1/sqrt(2) (B) sqrt(2) (C) 2 (D) none of these

If any tangent to the ellipse makes intercepts of lengths p and q on the axes, prove that (a^(2))/(p^(2))+(b^(2))/(q^(2))=1

If AM and GM of x and y are in the ratio p: q, then x: y is : a) p - sqrt(p^(2)+ q^(2)) : p +sqrt (p^(2)+ q^(2)) b) p +sqrt(p^(2) - q^ (2) ) : p- sqrt(p^(2)-q^(2)) c) p : q d) p + sqrt(p^(2) + q^(2)) : p-sqrt(p^(2) + q^(2))

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.

sqrt(x^(p-q))sqrt(x^(q-r))sqrt(x^(r-p))=1