The line `3x+6y=k` intersects the curve `2x^2+3y^2=1` at points `A and B` . The circle on `A B` as diameter passes through the origin. Then the value of `k^2` is__________

The line x =2 y intersects the ellipse (x^(2))/(4) +y^(2) = 1 at the points P and Q . The equation of the circle with pq as diameter is _

For the curve y = 4x^(3) – 2x^(5) , find all the points at which the tangent passes through the origin.

For the curve y=4x^3-2x^5, find all the points at which the tangent passes through the origin.

Show that the circle with the portion of the line 3x + 4y = 12 intercepted between the axes as diameter, passes through the origin.

Find the locus of the mid points of the chords of the circle x^2 + y^2 -2x -6y - 10 = 0 which pass through the origin.

A line L passing through the point (2, 1) intersects the curve 4x^2+y^2-x+4y-2=0 at the point Aa n dB . If the lines joining the origin and the points A ,B are such that the coordinate axes are the bisectors between them, then find the equation of line Ldot

The normal to-be circle x^2 +y^2 - 4x + 6y -12=0 passes through the point

The line y = mx intersects the circle x^(2)+y^(2) -2x - 2y = 0 and x^(2)+y^(2) +6x - 8y =0 at point A and B (points being other than origin). The range of m such that origin divides AB internally is

If the line y=2x+k is a tangent to the curve x^(2)=4y , then the value of k is -

x+y=6 and x+2y=4 are two diameters of a circle. If the circel passes through the point (6,2), then its radius is-