Straight Lines

The value of c for which line y = 2x + c is tangent to circle x² + y² = (√5)² is

Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point:

The point P(a,b) undergoes the following three transformations successively :
(a) reflection about the line y = x
(b) translation through 2 units along the positive direction of x-axis.
(c) rotation through angle π/4 about the origin in the anti-clockwise direction.
If the co-ordinates of the final position of the point P are (-1/√2, 7/√2), then the value of 2a + b is equal to :

In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x +y = 4. Let the point B lie on the line x + 3y = 7. If (α, β) is the centroid of ∆ABC, then 15(α + β) is equal to:

Let dy/dx = (ax-by+a) / (bx+cy+a), where a, b, c, are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the point (11, 6) from this circle is:

Let a vector αi + βj be obtained by rotating the vector √3i + j by an angle 45° about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices (α, β), (0, β) and (0, 0) is equal to :

Let A(-1,1), B(3, 4) and C(2,0) be given three points. A line y = mx, m > 0, intersects lines AC and BC at point P and Q respectively. Let A? and A? be the area of ?ABC and ?PQC respectively, such that A? = 3A?, then the value of m is equal to:

Let the lengths of intercepts on x-axis and y-axis made by the circle x² + y² + ax + 2ay + c = 0, (a < 0) be 22 and 25, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line x + 2y = 0, is equal to :

Let L be a tangent line to parabola y² = 4x - 20 at (6,2). If L is also a tangent to the ellipse x²/2 + y²/b = 1, then the value of b is equal to:

The solutions of the equation
|1+sin²x, sin²x, sin²x|
|cos²x, 1+cos²x, cos²x| = 0, (0 < x < π), are:
|4sin2x, 4sin2x, 1+4sin2x|