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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...
##### 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...
##### 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...
##### 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...
##### 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 ...
##### 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}+...
##### 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)\$...
##### 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=...
##### \$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...
##### 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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