If the circles x2 + y2 + 5Kx + 2y + K = 0 and 2(x2 + y2) + 2Kx + 3y –1 = 0, (K$$\in$$R), intersect at the points P and Q, then the line 4x + 5y – K = 0 passes through P and Q, for :

A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (–8, 5) and (6, 5), then the area of the rectangle (in sq. units) is :

The common tangent to the circles x 2 + y2 = 4 and x2 + y2 + 6x + 8y – 24 = 0 also passes through the point :

The tangent and the normal lines at the point ( $$\sqrt{3}$$, 1) to the circle x2 + y2 = 4 and the x-axis form a triangle. The area of this triangle (in square units) is :

The sum of the squares of the lengths of the chords intercepted on the circle, x2 + y2 = 16, by the lines, x + y = n, n $$\in$$ N, where N is the set of all natural numbers, is :

Let C1 and C2 be the centres of the circles x2 + y2 – 2x – 2y – 2 = 0 and x2 + y2 – 6x – 6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC1QC2 is :

If a variable line, 3x + 4y – $$\lambda$$ = 0 is such that the two circles x2 + y2 – 2x – 2y + 1 = 0 and x2 + y2 – 18x – 2y + 78 = 0 are on its opposite sides, then the set of all values of $$\lambda$$ is the interval :

Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is :

A square is inscribed in the circle x2 + y2 – 6x + 8y – 103 = 0 with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is :

The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is :

If the area of an equilateral triangle inscribed in the circle x2 + y2 + 10x + 12y + c = 0 is $$27\sqrt{3}$$ sq units then c is equal to :

If a circle C passing through the point (4, 0) touches the circle x2 + y2 + 4x – 6y = 12 externally at the point (1, – 1), then the radius of C is :

If the circles x2 + y2 $$-$$ 16x $$-$$ 20y + 164 = r2 and (x $$-$$ 4)2 + (y $$-$$ 7)2 = 36 intersect at two distinct points, then :

Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then :

If a circle of radius R passes through the origin O and intersects the coordinates axes at A and B, then the locus of the foot of perpendicular from O on AB is :

If the normal to the ellipse 3x2 + 4y2 = 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent to the ellipse at P passes through Q(4,4) then PQ is equal to :

An ellipse, with foci at (0, 2) and (0, –2) and minor axis of length 4, passes through which of the following points?

The tangent and normal to the ellipse 3x2 + 5y2 = 32 at the point P(2, 2) meet the x-axis at Q and R, respectively. Then the area (in sq. units) of the triangle PQR is :

If the line x – 2y = 12 is tangent to the ellipse {{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1 at the point \left( {3, - {9 \over 2}} \right) , then the length of the latus rectum of the ellipse is :

If the tangent to the parabola y2 = x at a point ($$\alpha$$, $$\beta$$), ($$\beta$$ > 0) is also a tangent to the ellipse, x2 + 2y2 = 1, then $$\alpha$$ is equal to :