Let `C` be any circle with centre `(0,sqrt(2))dot` Prove that at most two rational points can be there on `Cdot` (A rational point is a point both of whose coordinates are rational numbers)

Let p,q be chosen one by one from the set {1, sqrt(2),sqrt(3), 2, e, pi} with replacement. Now a circle is drawn taking (p,q) as its centre. Then the probability that at the most two rational points exist on the circle is (rational points are those points whose both the coordinates are rational)

Prove that form any external point two tangents can be drawn to circle.

Prove that the two different circle can never intersect at more that two point .

Define a rational number and prove that sqrt(2) is not rational.

Statement 1: If in a triangle, orthocentre, circumcentre and centroid are rational points, then its vertices must also be rational points. Statement : 2 If the vertices of a triangle are rational points, then the centroid, circumcentre and orthocentre are also rational points.

Let n be the number of points having rational coordinates equidistant from the point (0,sqrt3) , then

The number of rational points on the line joining (sqrt(5), 3) and (3, sqrt(3)) is

Find the number of rational points on the ellipse (x^2)/9+(y^2)/4=1.

Prove that the maximum number of points with rational coordinates on a circle whose center is (sqrt(3),0) is two.

Find the equation of a curve passing through the point (0, -2) given that at any point (x,y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.