If the curve x2 + 2y2 = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is :

If the points of intersections of the ellipse {{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1 and the circle x2 + y2 = 4b, b > 4 lie on the curve y2 = 3x2, then b is equal to :

Let a tangent be drawn to the ellipse {{{x^2}} \over {27}} + {y^2} = 1 at $$(3\sqrt{3}\cos \theta ,\sin \theta )$$ where 0 \in \left( {0,{\pi \over 2}} \right). Then the value of $$\theta$$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :

Let {E_1}:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,a > b. Let E2 be another ellipse such that it touches the end points of major axis of E1 and the foci of E2 are the end points of minor axis of E1. If E1 and E2 have same eccentricities, then its value is :

Let an ellipse E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1, $$a^2>b^2$$, passes through \left( {\sqrt {{3 \over 2}} ,1} \right) and has eccentricity {1 \over {\sqrt 3 }}. If a circle, centered at focus F($$\alpha$$, 0), $$\alpha$$ > 0, of E and radius {2 \over {\sqrt 3 }}, intersects E at two points P and Q, then PQ2 is equal to :

If a tangent to the ellipse x2 + 4y2 = 4 meets the tangents at the extremities of it major axis at B and C, then the circle with BC as diameter passes through the point :

A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity {1 \over 3} and the distance of the nearer focus from this directrix is {8 \over {\sqrt {53} }}, then the equation of the other directrix can be :

On the ellipse {{{x^2}} \over 8} + {{{y^2}} \over 4} = 1 let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line x + 2y = 0. Let S and S' be the foci of the ellipse and e be its eccentricity. If A is the area of the triangle SPS' then, the value of (5 $$-$$ e2). A is :

If x2 + 9y2 $$-$$ 4x + 3 = 0, x, y $$\in$$ R, then x and y respectively lie in the intervals :

The line $$12x\cos \theta +5y\sin \theta =60$$ is tangent to which of the following curves?

The locus of mid-points of the line segments joining ($$-$$3, $$-$$5) and the points on the ellipse {{{x^2}} \over 4} + {{{y^2}} \over 9} = 1 is :

An angle of intersection of the curves, {{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1 and x2 + y2 = ab, a > b, is :

Let $$\theta$$ be the acute angle between the tangents to the ellipse {{{x^2}} \over 9} + {{{y^2}} \over 1} = 1 and the circle $$x^2+y^2=3$$ at their point of intersection in the first quadrant. Then tan$$\theta$$ is equal to :

Let R = {(P, Q) | P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1, $$-$$1) is the set :

The number of elements in the set {x $$\in$$ R : (|x| $$-$$ 3) |x + 4| = 6} is equal to :

Let A = {2, 3, 4, 5, ....., 30} and '$$\simeq$$' be an equivalence relation on A $$\times$$ A, defined by (a, b) $$\simeq$$ (c, d), if and only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to :

In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?

Define a relation R over a class of n $$\times$$ n real matrices A and B as "ARB iff there exists a non-singular matrix P such that PAP$$-$$1 = B". Then which of the following is true?

Let N be the set of natural numbers and a relation R on N be defined by $$R=\{(x,y)\in N\times N:x^3-3x^{2}y-x{y}^2+3y^3=0\}$$. Then the relation R is :

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :