Prove that the curve sin x=cos y does not intersect the circle `x^2+y^2=1` at real points.

Prove that the curve y = x^2 and xy = k intersect orthogonally if 8k^2 = 1 .

If the straight line represented by x cos alpha + y sin alpha = p intersect the circle x^2+y^2=a^2 at the points A and B , then show that the equation of the circle with AB as diameter is (x^2+y^2-a^2)-2p(xcos alpha+y sin alpha-p)=0

Prove that the curves "x"="y"^2 and "x y"="k" intersect at right angles if 8"k"^2=1.

Tangents are drawn to the circle x^(2)+y^(2)=16 at the points where it intersects the circle x^(2)+y^(2)-6x-8y-8=0 , then the point of intersection of these tangents is

Statement I The curve y=81^(sin^(2)x)+81^(cos^(2)x)-30 intersects X-axis at eight points in the region -pi le x le pi . Statement II The curve y=sinx or y=cos x intersects the X-axis at infinitely many points.

The locus of the centre of the circle passing through the intersection of the circles x^2+y^2= 1 and x^2 + y^2-2x+y=0 is

Find the point of intersection of the circle x^2+y^2-3x-4y+2=0 with the x-axis.

The equation of the circle passing through (1,2) and the points of intersection of the circles x^2+y^2-8x-6y+21=0 and x^2+y^2-2x-15=0 is

A variable circle which always touches the line x+y-2=0 at (1, 1) cuts the circle x^2+y^2+4x+5y-6=0 . Prove that all the common chords of intersection pass through a fixed point. Find that points.