Let `x^2 + y^2 = r^2` and `xy = 1` intersect at `A & B` in first quadrant, If `AB = sqrt14` then find the value of `r.`

Let x^(2)+y^(2)=r^(2) and xy=1 intersect at A&B in first quadrant,If AB=sqrt(14) then find the value of r.

Consider the curves C_(1):|z-2|=2+Re(z) and C_(2):|z|=3 (where z=x+iy,x,y in R and i=sqrt(-1) .They intersect at P and Q in the first and fourth quadrants respectively.Tangents to C_(1) at P and Q intersect the x- axis at R and tangents to C_(2) at P and Q intersect the x -axis at S .If the area of Delta QRS is lambda sqrt(2) ,then find the value of (lambda)/(2)

If the angle between x^2/9+y^2/1=1 and the circle x^2+y^2=3 at their point of intersection in 1st quadrant is theta then find value of tan theta

Consider the family of circles x^2 + y^2= r^2 2 lt r lt 5 . If in the first quadrant, the common tangent to a circle of this family and the ellipse frac{x^(2)}{25} +frac{y^(2)}{4} =1 meets the axes at A and B then find the equation of the locus of middle point of AB.

If x+y=12 and xy=14 find the values of x^(2)+y^(2)

If x+y=12 and xy=14, find the value of x^(2)+y^(2)

let A(x_(1),0) and B(x_(2),0) be the foci of the hyperbola (x^(2))/(9)-(y^(2))/(16)=1 suppose parabola having vertex at origin and focus at B intersect the hyperbola at P in first quadrant and at point Q in fourth quadrant.

Let A be the area of the region in the first quadrant bounded by the x-axis the line 2y = x and the ellipse (x^(2))/(9) + (y^(2))/(1) = 1 . Let A' be the area of the region in the first quadrant bounded by the y-axis the line y = kx and the ellipse . The value of 9k such that A and A' are equal is _____

Two circles centres A and B radii r_1 and r_2 respectively. (i) touch each other internally iff |r_1 - r_2| = AB . (ii) Intersect each other at two points iff |r_1 - r_2| ltAB lt r_1 r_2 . (iii) touch each other externally iff r_1 + r_2 = AB . (iv) are separated if AB gt r_1 + r_2 . Number of common tangents to the two circles in case (i), (ii), (iii) and (iv) are 1, 2, 3 and 4 respectively. If circles (x-1)^2 + (y-3)^2 = r^2 and x^2 + y^2 - 8x + 2y + 8=0 intersect each other at two different points, then : (A) 1ltrlt5 (B) 5ltrlt8 (C) 2ltrlt8 (D) none of these