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

The line x+2y+a=0 intersects the circle x^2+y^2-4=0 at two distinct points A and B. Another line 12x-6y-41=0 intersects the circle x^2+y^2-4x-2y+1=0 at two distincts points C and D. The value of 'a' so that the line x+2y+a=0 intersects the circle x^2+y^2-4=0 at two distinct points A and B is

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.

Given two circle x^(2)+y^(2)-6x-2y+5=0&x^(2)+y^(2)+6x+22y+5=0. The tangent at (2,-1) to the first circle passes outside the second circle intersects the second circle in 2 real the second circle intersects the second circle in 2 real points passes through the centre of the second circle

The line x cos alpha+y sin alpha=p, intersects the circle x^(2)+y^(2)=4 at A and B .If the chord AB makes an angle 30^(@) at a point on the circumference of the circle,then p^(2)=

Prove that the curve x^(2)+y^(2)+x+2y=0 and xy+2x-y=0 intersect orthogonally at the origin.