HARD
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If the sides of a triangle be along the lines a1x+b1y+c1=0, a2x+b2y+c2=0 and a3x+b3y+c3=0 where
ai, bi, ciQ,  then which of the following point(s) for given triangle must have necessarily rational coordinates?

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Important Questions on Special Points of a Triangle

MEDIUM
A point on the straight line, 3x+5y=15 which is equidistant from the coordinate axes will lie only in:
HARD
Let a and b be any two numbers satisfying 1a2+1b2=14. Then, the foot of perpendicular from the origin on the variable line xa+yb=1 lies on :
HARD
If 2x-y+1=0 is a tangent to the hyperbola x2a2-y216=1 , then which of the following CANNOT be sides of a right angled triangle?
EASY
The length of the perpendicular from the origin, on normal to the curve, x2+2xy-3y2=0, at the point 2,2, is.
HARD
If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is :
HARD
Let two points be A1,-1 and B0,2. If a point Px', y' be such that the area of PAB=5 sq. units and it lies on the line 3x+y-4λ=0, then a value of λ is
EASY
A straight line through a fixed point 2,3 intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:
MEDIUM

A ray of light coming from the point (2,23) is incident at an angle 30° on the line x=1 at the point A. The ray gets reflected on the line x=1 and meets x -axis at the point B. Then, the line AB passes through the point
 

MEDIUM
Two sides of a parallelogram are along the lines, x+y=3 and x-y+3=0. If its diagonals intersect at 2,4, then one of its vertex is:
EASY
A line has slope m and y-intercept 4. The distance between the origin and the line is equal to
EASY
The reflection of the point (α, β, γ) in the x y-plane is
HARD

For a>b>c>0, the distance between (1,1) and the point of intersection of the lines ax+by+c=0 and bx+ay+c=0 is less than 2 2 . Then

MEDIUM
The set of all possible values of θ in the interval 0,π for which the points 1, 2 and sinθ, cosθ lie on the same side of the line x+y=1 is?
EASY
Suppose that the points h,k,1,2 and -3,4 lie on the line L1. If a line L2 passing through the points h,k and 4,3 is perpendicular to L1, then kh equals:
MEDIUM
Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point-1, -4 in the line is :
EASY
The foot of the perpendicular drawn from the point (1, 8, 4) on the line joining the point (0, -11, 4) and (2,-3,1) is
EASY
The polar coordinate of a point P is 2, -π4. The polar coordinate of the point Q which is such that line joining PQ is bisected perpendicularly by the initial line, is
MEDIUM
Let C be the circle with centre 0, 0 and radius 3 unit. The equation of the locus of the mid points of the chords of the circle C that subtend an angle of 2π3 at its centre, is:
MEDIUM
If the sum of distances from a point P on two mutually perpendicular straight lines is 1 unit, then the locus of P is
HARD
The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices 0, 0, 0, 41 and 41, 0 is