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##### The point \$(2,3)\$ undergoes the following three transformations successively(i) reflection about the line \$y=x\$(ii) translation through a distance 2 units along the positive direction of \$y\$-axis(iii) rotation through an angle of \$45^{circ}\$ about the origin in the anti-clockwise direction. The final coordinates of the point are(A) \$left(frac{1}{sqrt{2}}, frac{7}{sqrt{2}}ight)\$(B) \$left(-frac{1}{sqrt{2}}, frac{7}{sqrt{2}}ight)\$(C) \$left(frac{1}{sqrt{2}},-frac{7}{sqrt{2}}ight)\$(D) none of these
The point \$(2,3)\$ undergoes the following three transformations successively (i) reflection about the line \$y=x\$ (ii) translation through a distance 2 units along the positive direction of \$y\$-axis (iii) rotation through an angle of \$45^{circ}\$ about the origin in the anti-clockwise direction. The f...
##### Lines \$L_{1}=a x+b y+c=0\$ and \$L_{2}=L x+m y+n=0\$intersect at the point \$P\$ and make an angle \$heta\$ with each other. The equation of line \$L\$ different from \$L_{2}\$ which passes through \$mathrm{P}\$ and makes the same angle \$heta\$ with \$L_{1}\$ is(A) \$2(a l+b m)(a x+b y+c)-left(a^{2}+b^{2}ight)(l x+m y+n)\$\$=0\$(B) \$2(a l+b m)(a x+b y+c)+left(a^{2}+b^{2}ight)(l x+m y+n)\$\$=0\$(C) \$2left(a^{2}+b^{2}ight)(a x+b y+c)-(a l+b m)(l x+m y+n)\$\$=0\$(D) none of these
Lines \$L_{1}=a x+b y+c=0\$ and \$L_{2}=L x+m y+n=0\$ intersect at the point \$P\$ and make an angle \$ heta\$ with each other. The equation of line \$L\$ different from \$L_{2}\$ which passes through \$mathrm{P}\$ and makes the same angle \$ heta\$ with \$L_{1}\$ is (A) \$2(a l+b m)(a x+b y+c)-left(a^{2}+b^{2} ight)(...
##### The equations of the perpendicular bisector of the sides \$A B\$ and \$A C\$ of a \$Delta A B C\$ are \$x-y+5=0\$ and \$x+\$ \$2 y=0\$, respectively. If the point \$A\$ is \$(1,-2)\$ then the equation of the line \$B C\$ is(A) \$14 x+23 y=40\$(B) \$14 x-23 y=40\$(C) \$23 x+14 y=40\$(D) \$23 x-14 y=40\$
The equations of the perpendicular bisector of the sides \$A B\$ and \$A C\$ of a \$Delta A B C\$ are \$x-y+5=0\$ and \$x+\$ \$2 y=0\$, respectively. If the point \$A\$ is \$(1,-2)\$ then the equation of the line \$B C\$ is (A) \$14 x+23 y=40\$ (B) \$14 x-23 y=40\$ (C) \$23 x+14 y=40\$ (D) \$23 x-14 y=40\$...
##### 1. The equation of a family of lines is given by \$(2+3 t)\$ \$x+(1-2 t) y+4=0\$, where \$t\$ is the parameter. The equation of a straight line, belonging to this family, at the maximum distance from the point \$(2,3)\$ is(A) \$21 x+14 y=0\$(B) \$21 x-14 y=0\$(C) \$14 x-21 y=0\$(D) none of these
1. The equation of a family of lines is given by \$(2+3 t)\$ \$x+(1-2 t) y+4=0\$, where \$t\$ is the parameter. The equation of a straight line, belonging to this family, at the maximum distance from the point \$(2,3)\$ is (A) \$21 x+14 y=0\$ (B) \$21 x-14 y=0\$ (C) \$14 x-21 y=0\$ (D) none of these...
##### \$A B C D\$ is a square whose vertices \$A, B, C\$ and \$D\$ are \$(0,0),(2,0),(2,2)\$ and \$(0,2)\$, respectively. This square is rotated in the \$X-Y\$ plane with an angle of \$30^{circ}\$ in anti-clockwise direction about an axis passing through the vertex \$A\$. The equation of the diagonal \$B D\$ of this rotated square is(A) \$sqrt{3} x+(1-sqrt{3}) y=sqrt{3}\$(B) \$(1+sqrt{3}) x-(1-sqrt{2})=2\$(C) \$(2-sqrt{3}) x+y=2(sqrt{3}-1)\$(D) none of these
\$A B C D\$ is a square whose vertices \$A, B, C\$ and \$D\$ are \$(0,0),(2,0),(2,2)\$ and \$(0,2)\$, respectively. This square is rotated in the \$X-Y\$ plane with an angle of \$30^{circ}\$ in anti-clockwise direction about an axis passing through the vertex \$A\$. The equation of the diagonal \$B D\$ of this rotate...
##### The equations of the straight lines passing through \$(-2,-7)\$ and cutting an intercept of length three units between the straight lines \$4 x+3 y=12\$ and \$4 x+3 y=\$ 3 are(A) \$x+2=0, y+7=frac{7}{24}(x+2)\$(B) \$x-2=0, y+7=-frac{7}{24}(x+2)\$(C) \$x+2=0, y+7=-frac{7}{24}(x+2)\$(D) \$x+2=0, y+7=-frac{7}{12}(x+2)\$
The equations of the straight lines passing through \$(-2,-7)\$ and cutting an intercept of length three units between the straight lines \$4 x+3 y=12\$ and \$4 x+3 y=\$ 3 are (A) \$x+2=0, y+7=frac{7}{24}(x+2)\$ (B) \$x-2=0, y+7=-frac{7}{24}(x+2)\$ (C) \$x+2=0, y+7=-frac{7}{24}(x+2)\$ (D) \$x+2=0, y+7=-frac{7}{1...
##### The coordinates of the point which is at unit distance from the lines \$L_{1} equiv 3 x-4 y+1=0\$ and \$L_{2} equiv 8 x+6 y+\$ \$1=0\$ and lies below \$L_{1}\$ and above \$L_{2}\$ are(A) \$left(frac{6}{5}, frac{1}{10}ight)\$(B) \$left(frac{6}{5},-frac{1}{10}ight)\$(C) \$left(frac{6}{5}, frac{1}{5}ight)\$(D) \$left(frac{6}{5},-frac{1}{5}ight)\$
The coordinates of the point which is at unit distance from the lines \$L_{1} equiv 3 x-4 y+1=0\$ and \$L_{2} equiv 8 x+6 y+\$ \$1=0\$ and lies below \$L_{1}\$ and above \$L_{2}\$ are (A) \$left(frac{6}{5}, frac{1}{10} ight)\$ (B) \$left(frac{6}{5},-frac{1}{10} ight)\$ (C) \$left(frac{6}{5}, frac{1}{5} ight)\$ (D) ...
##### The vertices of a triangle are \$Aleft(x_{1}, x_{1} an alphaight), Bleft(x_{2}, x_{2}ight.\$\$an eta\$ ) and \$Cleft(x_{3}, x_{3} an gammaight)\$. If the circumcentre of triangle \$A B C\$ coincides with the origin and \$H(a, b)\$ be its orthocentre then \$frac{a}{h}=\$(A) \$frac{cos alpha+cos eta+cos gamma}{cos alpha cdot cos eta cdot cos gamma}\$(B) \$frac{sin alpha+sin eta+sin gamma}{sin alpha cdot sin eta cdot sin gamma}\$(C) \$frac{an alpha+an eta+an gamma}{an alpha cdot an eta cdot an gamma}\$(D) \$frac{cos
The vertices of a triangle are \$Aleft(x_{1}, x_{1} an alpha ight), Bleft(x_{2}, x_{2} ight.\$ \$ an eta\$ ) and \$Cleft(x_{3}, x_{3} an gamma ight)\$. If the circumcentre of triangle \$A B C\$ coincides with the origin and \$H(a, b)\$ be its orthocentre then \$frac{a}{h}=\$ (A) \$frac{cos alpha+cos eta+cos ...
##### \$O X\$ and \$O Y\$ are two coordinate axes. On \$O Y\$ is taken a fixed point \$P\$ and on \$O X\$ any point \$Q .\$ On \$P Q\$ an equilateral triangle is described, its vertex \$R\$ being on the side of \$P Q\$ away from \$O\$, then the locus of \$R\$ will be(A) straight line(B) circle(C) ellipse(D) parabola
\$O X\$ and \$O Y\$ are two coordinate axes. On \$O Y\$ is taken a fixed point \$P\$ and on \$O X\$ any point \$Q .\$ On \$P Q\$ an equilateral triangle is described, its vertex \$R\$ being on the side of \$P Q\$ away from \$O\$, then the locus of \$R\$ will be (A) straight line (B) circle (C) ellipse (D) parabola...
##### If the vertices of a variable triangle are \$(3,4)\$, ( \$5 mathrm{cos}\$ \$heta, 5 sin heta\$ ) and \$(5 sin heta,-5 cos heta)\$, then the locus of itsorthocentre is(A) \$(x+y-1)^{2}+(x-y-7)^{2}=100\$(B) \$(x+y-7)^{2}+(x-y+1)^{2}=100\$(C) \$(x+y-7)^{2}+(x-y-1)^{2}=100\$(D) \$(x+y+7)^{2}+(x+y-1)^{2}=100\$
If the vertices of a variable triangle are \$(3,4)\$, ( \$5 mathrm{cos}\$ \$ heta, 5 sin heta\$ ) and \$(5 sin heta,-5 cos heta)\$, then the locus of its orthocentre is (A) \$(x+y-1)^{2}+(x-y-7)^{2}=100\$ (B) \$(x+y-7)^{2}+(x-y+1)^{2}=100\$ (C) \$(x+y-7)^{2}+(x-y-1)^{2}=100\$ (D) \$(x+y+7)^{2}+(x+y-1)^{2}=100\$...

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