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If the point $Pleft(a^{2}, aight.$ ) lies in the region corresponding to the acute angle between the lines $2 y=x$ and $4 y=x$, then(A) $a in(2,6)$(B) $a in(4,6)$(C) $a in(2,4)$(D) none of these
If the point $Pleft(a^{2}, a ight.$ ) lies in the region corresponding to the acute angle between the lines $2 y=x$ and $4 y=x$, then (A) $a in(2,6)$ (B) $a in(4,6)$ (C) $a in(2,4)$ (D) none of these...
5 answers
The point $(4,1)$ undergoes the following three successive transformations(A) Reflection about the line $y=x-1$(B) Translation through a distance 1 unit along the positive $x$-axis(C) Rotation through an angle $frac{pi}{4}$ about the origin in the anti-clockwise direction. Then, the coordinates of the final point are(A) $(4,3)$(B) $left(frac{7}{2}, frac{7}{2}ight)$(C) $(0,3 sqrt{2})$(D) $(3,4)$
The point $(4,1)$ undergoes the following three successive transformations (A) Reflection about the line $y=x-1$ (B) Translation through a distance 1 unit along the positive $x$-axis (C) Rotation through an angle $frac{pi}{4}$ about the origin in the anti-clockwise direction. Then, the coordinates o...
5 answers
A light ray emerging from the point source placed at $P(2,3)$ is reflected at point ' $heta$ on the $y$-axis and then passes through the point $R(5,10)$. Coordinates of ' $Q$ ' are(A) $(0,3)$(B) $(0,2)$(C) $(0,5)$(D) none of these
A light ray emerging from the point source placed at $P(2,3)$ is reflected at point ' $ heta$ on the $y$-axis and then passes through the point $R(5,10)$. Coordinates of ' $Q$ ' are (A) $(0,3)$ (B) $(0,2)$ (C) $(0,5)$ (D) none of these...
5 answers
The distance between two parallel lines is unity. A point $P$ lies between the lines at a distance $a$ from one of them. The length of a side of an equilateral triangle $P Q R$, vertex $Q$ of which lies on one of the parallel lines and vertex $R$ lies on the other line, is(A) $frac{2}{sqrt{3}} cdot sqrt{a^{2}+a+1}$(B) $frac{2}{sqrt{3}} sqrt{a^{2}-a+1}$(C) $frac{1}{sqrt{3}} sqrt{a^{2}+a+1}$(D) $frac{1}{sqrt{3}} sqrt{a^{2}-a+1}$
The distance between two parallel lines is unity. A point $P$ lies between the lines at a distance $a$ from one of them. The length of a side of an equilateral triangle $P Q R$, vertex $Q$ of which lies on one of the parallel lines and vertex $R$ lies on the other line, is (A) $frac{2}{sqrt{3}} cdot...
5 answers
Two points $A$ and $B$ are given. $P$ is a moving point on one side of the line $A B$ such that $angle P A B-angle P B A$ is a positive constant $2 heta$. The locus of the point $P$ is(A) $x^{2}+y^{2}+2 x y cot 2 heta=a^{2}$(B) $x^{2}+y^{2}-2 x y cot 2 heta=a^{2}$(C) $x^{2}+y^{2}+2 x y an 2 heta=a^{2}$(D) $x^{2}-y^{2}+2 x y cot 2 heta=a^{2}$.
Two points $A$ and $B$ are given. $P$ is a moving point on one side of the line $A B$ such that $angle P A B-angle P B A$ is a positive constant $2 heta$. The locus of the point $P$ is (A) $x^{2}+y^{2}+2 x y cot 2 heta=a^{2}$ (B) $x^{2}+y^{2}-2 x y cot 2 heta=a^{2}$ (C) $x^{2}+y^{2}+2 x y an 2 ...
5 answers
The four points $A(p, 0), B(q, 0), C(r, 0)$ and $D(s, 0)$ are such that $p, q$ are the roots of the equation $a x^{2}+2 h x+$ $b=0$ and $r, s$ are those of equation $a^{prime} x^{2}+2 h^{prime} x+b^{prime}=0$. If the sum of the ratios in which $C$ and $D$ divide $A B$ is zero, then(A) $a b^{prime}+a^{prime} b=2 h h^{prime}$(B) $a b^{prime}+a^{prime} b=h h^{prime}$(C) $a b^{prime}-a^{prime} b=2 h h^{prime}$(D) none of these
The four points $A(p, 0), B(q, 0), C(r, 0)$ and $D(s, 0)$ are such that $p, q$ are the roots of the equation $a x^{2}+2 h x+$ $b=0$ and $r, s$ are those of equation $a^{prime} x^{2}+2 h^{prime} x+b^{prime}=0$. If the sum of the ratios in which $C$ and $D$ divide $A B$ is zero, then (A) $a b^{prime}+...
5 answers
The coordinates of a point $P$ on the line $3 x+2 y+10$ $=0$ such that $|P A-P B|$ is maximum where $A$ is $(4,2)$ and $B$ is $(2,4)$, are(A) $(22,28)$(B) $(22,-28)$(C) $(-22,28)$(D) $(-22,-28)$
The coordinates of a point $P$ on the line $3 x+2 y+10$ $=0$ such that $|P A-P B|$ is maximum where $A$ is $(4,2)$ and $B$ is $(2,4)$, are (A) $(22,28)$ (B) $(22,-28)$ (C) $(-22,28)$ (D) $(-22,-28)$...
5 answers
A line through $A(-5,-4)$ meets the lines $x+3 y+2=0$, $2 x+y+4=0$ and $x-y-5=0$ at the point $B, C$ and $D$, respectively. If $left(frac{15}{A B}ight)^{2}+left(frac{10}{A C}ight)^{2}=left(frac{6}{A D}ight)^{2}$, the equa-tion of the line is(A) $2 x+3 y+22=0$(B) $2 x-3 y+22=0$(C) $3 x+2 y+22=0$(D) $3 x-2 y+22=0$
A line through $A(-5,-4)$ meets the lines $x+3 y+2=0$, $2 x+y+4=0$ and $x-y-5=0$ at the point $B, C$ and $D$, respectively. If $left(frac{15}{A B} ight)^{2}+left(frac{10}{A C} ight)^{2}=left(frac{6}{A D} ight)^{2}$, the equa- tion of the line is (A) $2 x+3 y+22=0$ (B) $2 x-3 y+22=0$ (C) $3 x+2 y+22=...
5 answers
$A(0,0), B(2,1)$ and $C(3,0)$ are the vertices of a $riangle A B C$ and $B D$ is its altitude. If the line through $D$ parallel to the side $A B$ intersects the side $B C$ at a point $K$ then the product of the areas of the triangles $A B C$ and $B D K$ is(A) 1(B) $frac{1}{2}$(C) $frac{1}{4}$(D) none of these
$A(0,0), B(2,1)$ and $C(3,0)$ are the vertices of a $ riangle A B C$ and $B D$ is its altitude. If the line through $D$ parallel to the side $A B$ intersects the side $B C$ at a point $K$ then the product of the areas of the triangles $A B C$ and $B D K$ is (A) 1 (B) $frac{1}{2}$ (C) $frac{1}{4}$ (D...
5 answers
A line cuts the $x$-axis at $A(7,0)$ and $y$-axis at $B(0,-5)$. A variable line $P Q$ is drawn $perp$ to $A B$ cutting the $x$-axis in $P$ and the $y$-axis in $Q .$ If $A Q$ and $B P$ intersect at $R$, then the locus of $R$ is(A) $x(x-7)+y(y+5)=0$(B) $x(x-7)-y(y+5)=0$(C) $x(x+7)+y(y-5)=0$(D) none of these
A line cuts the $x$-axis at $A(7,0)$ and $y$-axis at $B(0,-5)$. A variable line $P Q$ is drawn $perp$ to $A B$ cutting the $x$-axis in $P$ and the $y$-axis in $Q .$ If $A Q$ and $B P$ intersect at $R$, then the locus of $R$ is (A) $x(x-7)+y(y+5)=0$ (B) $x(x-7)-y(y+5)=0$ (C) $x(x+7)+y(y-5)=0$ (D) non...

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