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)

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)

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

A(1/(sqrt(2)),1/(sqrt(2))) is a point on the circle x^2+y^2=1 and B is another point on the circle such that are length A B=pi/2 units. Then, the coordinates of B can be

Draw a circle P and radius 3.4cm. Take point Q at a distance 5.5 cm. From the centre. Construct tangents to the circle from point Q.

Tangent drawn from the point P(4,0) to the circle x^2+y^2=8 touches it at the point A in the first quadrant. Find the coordinates of another point B on the circle such that A B=4 .

If the x-coordinate of a point P on the join of Q(22,1)a n dR(5,1,-2)i s4, then find its z- coordinate.

Draw a circle centre O . Take two points P and Q on the circle with such that angleAOB=120^(@). Draw tangents at points A and B.

A particle from the point P(sqrt3,1) moves on the circle x^2 +y^2=4 and after covering a quarter of the circle leaves it tangentially. The equation of a line along with the point moves after leaving the circle is

There are some experiment in which the outcomes cannot be identified discretely. For example, an ellipse of eccentricity 2sqrt(2)//3 is inscribed in a circle and a point within the circle is chosen at random. Now, we want to find the probability that this point lies outside the ellipse. Then, the point must lie in the shaded region shown in Figure. Let the radius of the circle be a and length of minor axis of the ellipse be 2b. Given that 1 - (b^(2))/(a^(2)) = (8)/(9) or (b^(2))/(a^(2)) = (1)/(9) Then, the area of circle serves as sample space and area of the shaded region represents the area for favorable cases. Then, required probability is p= ("Area of shaded region")/("Area of circle") =(pia^(2) - piab)/(pia^(2)) = 1 - (b)/(a) = 1 - (1)/(3) = (2)/(3) Now, answer the following questions. A point is selected at random inside a circle. The probability that the point is closer to the center of the circle than to its circumference is