Let `O` be the origin. If `A(1,0)a n dB(0,1)a n dP(x , y)` are points such that `x y >0a n dx+y<1,` then `P`

Consider the points O(0,0), A(0,1) , and B(1,1) in the x-y plane. Suppose that points C(x,1) and D(1,y) are chosen such that 0ltxlt1 . And such that O,C, and D are collinear. Let the sum of the area of triangles OAC and BCD be denoted by S. Then which of the following is/are correct?.

If the image of the co-ordinate of a point (x_1,y_1) is (x_2,y_2) with respect to the straight line lx+my+n=0,then show that (x_2-x_1)/l=(y_2-y_1)/m=(-2(lx_1+my_1+n))/(l^2+m^2)

A line L passing through the point (2, 1) intersects the curve 4x^2+y^2-x+4y-2=0 at the point Aa n dB . If the lines joining the origin and the points A ,B are such that the coordinate axes are the bisectors between them, then find the equation of line Ldot

Let (x , y) be such that sin^(-1)(a x)+cos^(-1)(y)+cos^(-1)(b x y)=pi/2dot Match the statements in column I with statements in column II Ifa=1a n db=0,t h e n(x , y) , (p) lies on the circle x^2+y^2=1 Ifa=1a n db=1,t h e n(x , y) , (q) l i e son(x^2-1)(y^2-1)=0 Ifa=1a n db=2,t h e n(x , y) , (r) lies on y=x Ifa=2a n db=2,t h e n(x , y) , (s) lies on (4x^2-1)(y^2-1)=0

A figure is bounded by the curves y=x^2+1, y=0,x=0,a n dx=1. At what point (a , b) should a tangent be drawn to curve y=x^2+1 for it to cut off a trapezium of greatest area from the figure?

If O is the origin and O Pa n dO Q are the tangents from the origin to the circle x^2+y^2-6x+4y+8-0 , then the circumcenter of triangle O P Q is (3,-2) (b) (3/2,-1) (3/4,-1/2) (d) (-3/2,1)

If O Aa n dO B are equal perpendicular chords of the circles x^2+y^2-2x+4y=0 , then the equations of O Aa n dO B are, where O is the origin. (a) 3x+y=0 and 3x-y=0 (b) 3x+y=0 and 3y-x=0 (c) x+3y=0 and y-3x=0 (d) x+y=0 and x-y=0

Let f(x)=sqrt(|x|-{x})(w h e r e{dot} denotes the fractional part of (x)a n dX , Y are its domain and range, respectively). Then (a) X in (-oo,1/2) and Y in (1/2,oo) (b) X in (-oo in ,1/2)uu[0,oo)a n dY in (1/2,oo) (c) X in (-oo,-1/2)uu[0,oo)a n dY in [0,oo) (d) none of these

Let (x,y) be any point on the parabola y^2 = 4x . Let P be the point that divides the line segment from (0,0) and (x,y) in the ratio 1:3. Then the locus of P is :

For points P-=(x_1y_1) and Q-=(x_2,y_2) of the coordinate plane, a new distance d(P,Q) is defined by d (P,Q)=|X_1-X_2|+|y_1-y_2| Let O=(0,0),A=(1,2), B-=(2,3) and C-=(4,3) are four fixed points on x-y plane Answer the following questions based on above passage: Let R (x,y), such that R is equidistant from the points O and A with respect to new distance and if 0 < x < 1 and 0 < y < 2 then R lie on a line segment whose equation is