Let 0 < α < π/2 be a fixed angle. If P = (cos θ, sin θ) and Q = (cos (α – θ), sin (α – θ) then Q is obtained from P by
Clockwise rotation around the origin through an angle α
OP makes an angle θ with the positive direction of x-axis and OQ makes an angle (α – θ) with the positive direction of x-axis.
So that ∠POQ = α and thus Q is obtained from P by clockwise rotation through an angle α around the origin.
If the circumcentre of a triangle lies at the origin and centroid is the middle point of the line joining the points (a2 + 1, a2 + 1) and (2a, –2a), then the orthocenter lies on the line.
If a, b, c are unequal and different from 1 such that the points are collinear, then
If two vertices of a triangle are (–2, 3) and (5, –1), orthocentre lies at the origin and centroid on the line x + y = 7, then the third vertex lies at
The line L has intercepts a and b on the coordinate axes. The coordinate axes are rotated through a fixed angle, keeping the origin fixed. If p andq are the intercepts of the line L on the new axes, then
If P is a point (x, y) on the line, y = –3x such that P and the point (3, 4) are on the opposite sides of the line 3x – 4y = 8, then
The area enclosed by 2|x| + 3|y| ≤ 6 is
Let O be the origin, A (1, 0) and B (0, 1) and P (x, y) are points such thatxy > 0 and x + y < 1, then
If a line joining two points A (2, 0) and B (3, 1) is rotated about A in anticlockwise direction through and angle 15o, then equation of the line is the new position is
An equation of a line through the point (1, 2) whose distance from the point (3, 1) has the greatest value is
On the portion of the straight line x + y = 2 which is intercepted between the axes, a square is constructed, away from the origin, with this portion as one of its side. If p denotes the perpendicular distance of a side of this square from the origin, then the maximum value of p is