Let a variable line intersects the coordinate axis at points A & B, such that area of triangle AOB is always 2 square units. Then the line always touch the hyperbola. (A) xy=1 (B) xy=2 (C) xy = 4 (D) `x^2-y^2 = 1`

Prove that the straight line y=mx+2c sqrt(-m),m in R, always touches the hyperbola xy=c^(2)

A variable straight line passes through a fixed point (a, b) intersecting the co-ordinates axes at A & B. If 'O' is the origin, then the locus of the centroid of the triangle OAB is (A) bx+ ay-3xy=0 (B) bx+ay-2xy=0 (C) ax+by-3xy=0 (D) ax+by-2xy=0.

If a variable line has its intercepts on the coordinate axes e and e', where (e)/(2) and (e')/(2) are the eccentricities of a hyperbola and its conjugate hyperbola,then the line always touches the circle x^(2)+y^(2)=r^(2), where r=1 (b) 2 (c) 3 (d) cannot be decided

A variable straight line with slope 2 intersects the hyperbola xy=1 at two pooints.Find the locus of the mid points of the line segment.

A circle of radius 1 unit touches the positive x-axis and the positive y-axis at A and B , respectively. A variable line passing through the origin intersects the circle at two points D and E . If the area of triangle D E B is maximum when the slope of the line is m , then find the value of m^(-2)

A circle of radius 1 unit touches the positive x-axis and the positive y-axis at A and B , respectively. A variable line passing through the origin intersects the circle at two points D and E . If the area of triangle D E B is maximum when the slope of the line is m , then find the value of m^(-2)

A variable straight line of slope 4 intersects the hyperbola xy=1 at two points. Then the locus of the point which divides the line segment between these two points in the ratio 1:2 , is (A) x^2 + 16y^2 + 10xy+2=0 (B) x^2 + 16y^2 + 2xy-10=0 (C) x^2 + 16y^2 - 8xy + 19 = 0 (D) 16x^2 + y^2 + 10xy-2=0