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Number of equilateral triangles with $y=sqrt{3}(x-1)+2$ and $y=-sqrt{3} x$ as two of its sides, is (A) 0 (B) 1 (C) 2 (D) none of these...

If the distance of any point $P(x, y)$ from the origin is defined as $d(x, y)=operatorname{Max} .{|x|,|y|}$ and $d(x, y)=k$ (nonzero constant), then the locus of the point $P$ is (A) a straight line (B) a circle (C) a parabola (D) none of these...

If $a, b, c$ form an A. P. with common difference $d( eq 0)$ and $x, y, z$ form a G. P. with common ratio $r eq 1$ ), then the area of the triangle with vertices $(a, x),(b, y)$ and $(c, z)$ is independent of (A) $b$ (B) $r$ (C) $d$ (D) $x$...

A line of fixed length 2 units moves so that its ends are on the positive $x$-axis and that part of the line $x+y=$ 0 which lies in the second quadrant. The locus of the mid-point of the line has the equation (A) $(x+2 y)^{2}+y^{2}=1$ (B) $(x-2 y)^{2}+y^{2}=1$ (C) $(x+2 y)^{2}-y^{2}=1$ (D) none of t...

A straight line through the origin $O$ meets the parallel lines $4 x+2 y=9$ and $2 x+y+6=0$ at points $P$ and $Q$, respectively. The point $O$ divides the segment $P Q$ in the ratio (A) $1: 2$ (B) $3: 4$ (C) $2: 1$ (D) $4: 3$...

Let $O$ be the origin and let $A(2,0), B(0,2)$ be two points. If $P(x, y)$ is a point such that $x y>0$ and $x+y<$ 2 , then (A) $P$ lies either inside the triangle $O A B$ or in the third quadrant (B) $P$ cannot be inside the triangle $O A B$ (C) $P$ lies inside the triangle $O A B$ (D) none o...

Consider the equation $y-y_{1}=mleft(x-x_{1} ight)$. In this equation, if $m$ and $x_{1}$ are fixed and different lines are drawn for different values of $y^{1}$, then, (A) the lines will pass through a single point (B) there will be one possible line only (C) there will be a set of parallel lines (...

$D$ is a point on $A C$ of the triangle with vertices $A(2,$, 3), $B(1,-3), C(-4,-7)$ and $B D$ divides $A B C$ into two triangles of equal area. The equation of the line drawn through $B$ at right angles to $B D$ is (A) $y-2 x+5=0$ (B) $2 y-x+5=0$ (C) $y+2 x-5=0$ (D) $2 y+x-5=0$...

If two points $A(a, 0)$ and $B(-a, 0)$ are stationary and if $angle A-angle B= heta$ in $Delta A B C$, the locus of $C$ is (A) $x^{2}+y^{2}+2 x y an heta=a^{2}$ (B) $x^{2}-y^{2}+2 x y an heta=a^{2}$ (C) $x^{2}+y^{2}+2 x y cot heta=a^{2}$ (D) $x^{2}-y^{2}+2 x y cot heta=a^{2}$...

The straight line $y=x-2$ rotates about a point where it cuts the $x$-axis and becomes perpendicular to the straight line $a x+b y+c=0 .$ Then, its equation is (A) $a x+b y+2 a=0$ (B) $a x-b y-2 a=0$ (C) $b y+a y-2 b=0$ (D) $a y-b x+2 b=0$...