11. Straight Lines(11th)

SECTION A: Straight Lines

1. If the orthocentre of the triangle, whose vertices are $(1,2),(2,3)$ and $(3,1)$ is $(\alpha, \beta)$ , then the quadratic equation whose roots are $\alpha+4 \beta$ and $4 \alpha+\beta$ , is:

(A) $x^{2}-20 x+99=0$

(B) $x^{2}-22 x+120=0$

(D) $x^{2}-18 x+80=0$

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2. Let $B$ and $C$ be the two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y-2 x=2$ such that $\triangle A B C$ is an equilateral triangle. Then, the area of the $\triangle A B C$ is:

(A) $\frac{10}{\sqrt{3}}$

(B) $2 \sqrt{3}$

(D) $\frac{8}{\sqrt{3}}$

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3. A light ray emits from the origin making an angle $30^{\circ}$ with the positive $x$ -axis. After getting reflected by the line $x+y=1$ , if this ray intersects $x$ -axis at $Q$ , then the abscissa of $Q$ is:

(A) $\frac{2}{(\sqrt{3}-1)}$

(B) $\frac{2}{3-\sqrt{3}}$

(D) $\frac{2}{3+\sqrt{3}}$

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4. Let $m_{1}, m_{2}$ be the slopes of two adjacent sides of a square of side $a$ such that $a^{2}+11 a+3\left(m_{1}^{2}+m_{2}^{2}\right)=220$ . If one vertex of the square is $(10(\cos \alpha-\sin \alpha), 10(\sin \alpha+\cos \alpha))$ , where $\alpha \in\left(0, \frac{\pi}{2}\right)$ and the equation of one diagonal is $(\cos \alpha-\sin \alpha) x+(\sin \alpha+\cos \alpha) y=10$ , then $72\left(\sin ^{4} \alpha+\cos ^{4} \alpha\right)+a^{2}-3 a+13$ is equal to:

(A) 119

(B) 128

(D) 155

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5. Let $\mathrm{A}(\alpha,-2), \mathrm{B}(\alpha, 6)$ and $\mathrm{C}\left(\frac{\alpha}{4},-2\right)$ be vertices of a $\triangle \mathrm{ABC}$ . If $\left(5, \frac{\alpha}{4}\right)$ is the circumcentre of $\triangle \mathrm{ABC}$ , then which of the following is NOT correct about $\triangle \mathrm{ABC}$ ?

(A) area is 24

(B) perimeter is 25

(D) inradius is 2

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6. Let the circumcentre of a triangle with vertices $A(a, 3), B(b, 5)$ and $C(a, b), a b>0$ be $P(1,1)$ . If the line $AP$ intersects the line $BC$ at the point $\mathrm{Q}\left(k_{1}, k_{2}\right)$ , then $k_{1}+k_{2}$ is equal to:

(A) 2

(B) $\frac{4}{7}$

(D) 4

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7. The equations of the sides $\mathrm{AB}, \mathrm{BC}$ and $CA$ of a triangle $ABC$ are $2 x+y=0, x+\mathrm{p} y=39$ and $x-y=3$ respectively and $\mathrm{P}(2,3)$ is its circumcentre. Then which of the following is NOT true?

(A) $(\mathrm{AC})^{2}=9 \mathrm{p}$

(B) $(\mathrm{AC})^{2}+\mathrm{p}^{2}=136$

(D) $34<$ area $(\triangle \mathrm{ABC})<38$

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8. Let $A(1,1), B(-4,3), C(-2,-5)$ be vertices of a triangle $A B C, P$ be a point on side $B C$ , and $\Delta_{1}$ and $\Delta_{2}$ be the areas of triangles $A P B$ and $A B C$ , respectively. If $\Delta_{1}: \Delta_{2}=4: 7$ , then the area enclosed by the lines $A P, A C$ and the $x$ -axis is:

(A) $\frac{1}{4}$

(B) $\frac{3}{4}$

(D) 1

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9. A point $P$ moves so that the sum of squares of its distances from the points $(1,2)$ and $(-2,1)$ is 14. Let $f(x, y)=0$ be the locus of $P$ , which intersects the $x$ -axis at the points $\mathrm{A}, \mathrm{B}$ and the $y$ -axis at the points $\mathrm{C}, \mathrm{D}$ . Then the area of the quadrilateral $ACBD$ is equal to:

(A) $\frac{9}{2}$

(B) $\frac{3 \sqrt{17}}{2}$

(D) 9

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10. Let the point $P(\alpha, \beta)$ be at a unit distance from each of the two lines $L_{1}: 3 x-4 y+12=0$ , and $L_{2}: 8 x+6 y+11=0$ . If $P$ lies below $L_{1}$ and above $L_{2}$ , then $100(\alpha+\beta)$ is equal to:

(A) -14

(B) 42

(D) 14

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11. A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x-y-2=0$ at the point $B$ . If the length of the line segment $A B$ is $\frac{\sqrt{29}}{3}$ , then $B$ also lies on the line:

(A) $2 x+y=9$

(B) $3 x-2 y=7$

(D) $2 x-3 y=3$

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12. Let $\alpha_{1}, \alpha_{2}\left(\alpha_{1}<\alpha_{2}\right)$ be the values of $\alpha$ for the points $(\alpha,-3),(2,0)$ and $(1, \alpha)$ to be collinear. Then the equation of the line, passing through $\left(\alpha_{1}, \alpha_{2}\right)$ and making an angle of $\frac{\pi}{3}$ with the positive direction of the $x$ -axis, is:

(A) $x-\sqrt{3} y-3 \sqrt{3}+1=0$

(B) $\sqrt{3} x-y+\sqrt{3}+3=0$

(D) $\sqrt{3} x-y+\sqrt{3}-3=0$

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13. The distance of the origin from the centroid of the triangle whose two sides have the equations $x-2 y+1=0$ and $2 x-y-1=0$ and whose orthocenter is $\left(\frac{7}{3}, \frac{7}{3}\right)$ is:

(A) $\sqrt{2}$

(B) 2

(D) 4

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14. The distance between the two points $A$ and $A^{\prime}$ which lie on $y=2$ such that both the line segments $A B$ and $A^{\prime} B$ (where $B$ is the point $(2, 3)$ ) subtend angle $\frac{\pi}{4}$ at the origin, is equal to:

(A) 10

(B) $\frac{48}{5}$

(D) 3

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15. Let a triangle be bounded by the lines $L_{1}: 2 x+5 y=10 ; L_{2}:-4 x+3 y=12$ and the line $L_{3}$ , which passes through the point $P(2,3)$ , intersects $L_{2}$ at $A$ and $L_{1}$ at $B$ . If the point $P$ divides the line-segment $A B$ , internally in the ratio $1: 3$ , then the area of the triangle is equal to:

(A) $\frac{110}{13}$

(B) $\frac{132}{13}$

(D) $\frac{151}{13}$

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16. In an isosceles triangle $A B C$ , the vertex $A$ is $(6,1)$ and the equation of the base $B C$ is $2 x+y=4$ . Let the point $B$ lie on the line $x+3 y=7$ . If $(\alpha, \beta)$ is the centroid of $\triangle \mathrm{ABC}$ , then $15(\alpha+\beta)$ is equal to:

(A) 39

(B) 41

(D) 63

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17. Let $R$ be the point $(3,7)$ and let $P$ and $Q$ be two points on the line $x+y=5$ such that $P Q R$ is an equilateral triangle. Then the area of $\Delta P Q R$ is:

(A) $\frac{25}{4 \sqrt{3}}$

(B) $\frac{25 \sqrt{3}}{2}$

(D) $\frac{25}{2 \sqrt{3}}$

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18. Let the area of the triangle with vertices $\mathrm{A}(1, \alpha), \mathrm{B}(\alpha, 0)$ and $\mathrm{C}(0, \alpha)$ be 4 sq. units. If the points $(\alpha,-\alpha),(-\alpha, \alpha)$ and $\left(\alpha^{2}, \beta\right)$ are collinear, then $\beta$ is equal to:

(A) 64

(B) -8

(D) 512

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19. Let $A$ be the set of all points $(\alpha, \beta)$ such that the area of triangle formed by the points $(5,6),(3,2)$ and $(\alpha, \beta)$ is 12 square units. Then the least possible length of a line segment joining the origin to a point in $A$ , is:

(A) $\frac{4}{\sqrt{5}}$

(B) $\frac{16}{\sqrt{5}}$

(D) $\frac{12}{\sqrt{5}}$

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20. If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines, $x \operatorname{cosec} \alpha-y \sec \alpha=k \cot 2 \alpha$ and $x \sin \alpha+y \cos \alpha=k \sin 2 \alpha$ respectively, then $k^{2}$ is equal to:

(A) $4 p^{2}+q^{2}$

(B) $2 p^{2}+q^{2}$

(D) $p^{2}+4 q^{2}$

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