11. Straight Lines(11th)

SECTION A: Straight Lines

1. Let the equations of two sides of a triangle be $3 x-2 y+6=0$ and $4 x+5 y-20=0$ . If the orthocentre of this triangle is at $(1,1)$ , then the equation of its third side is:

(A) $122 y-26 x-1675=0$

(B) $122 y+26 x+1675=0$

(D) $26 x-122 y-1675=0$

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2. Consider the set of all lines $p x+q y+r=0$ such that $3 p+2 q+4 r=0$ . Which one of the following statements is true?

(A) The lines are not concurrent

(B) The lines are concurrent at the point $\left(\frac{3}{4}, \frac{1}{2}\right)$

(D) Each line passes through the origin

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3. A straight line through a fixed point $(2,3)$ intersects the coordinate axes at distinct points $P$ and $Q$ . If $O$ is the origin and the rectangle $OPRQ$ is completed, then the locus of $R$ is:

(A) $3 x+2 y=6 x y$

(B) $3 x+2 y=6$

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

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4. The foot of the perpendicular drawn from the origin, on the line, $3 x+y=\lambda(\lambda \neq 0)$ is $P$ . If the line meets $x$ -axis at $A$ and $y$ -axis at $B$ , then the ratio $\mathrm{BP}: \mathrm{PA}$ is:

(A) $1: 3$

(B) $3: 1$

(D) $9: 1$

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5. The sides of a rhombus $A B C D$ are parallel to the lines, $x-y+2=0$ and $7 x-y+3=0$ . If the diagonals of the rhombus intersect $P(1, 2)$ and the vertex $A$ (different from the origin) is on the $y$ -axis, then the coordinate of $A$ is:

(A) $\frac{5}{2}$

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

(D) $\frac{7}{2}$

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6. In a triangle $A B C$ , coordinates of $A$ are $(1,2)$ and the equations of the medians through $B$ and $C$ are respectively, $x+y=5$ and $x=4$ . Then area of $\triangle \mathrm{ABC}$ (in sq. units) is:

(A) 12

(B) 4

(D) 9

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7. A square, of each side 2, lies above the $x$ -axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle $30^{\circ}$ with the positive direction of the $x$ -axis, then the sum of the $x$ -coordinates of the vertices of the square is:

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

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

(D) $\sqrt{3}-1$

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8. Let $k$ be an integer such that the triangle with vertices $(k,-3 k),(5, k)$ and $(-k, 2)$ has area 28 sq. units. Then the orthocentre of this triangle is at the point:

(A) $\left(1, \frac{3}{4}\right)$

(B) $\left(1,-\frac{3}{4}\right)$

(D) $\left(2,-\frac{1}{2}\right)$

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9. A ray of light is incident along a line which meets another line, $7 x-y+1=0$ , at the point $(0,1)$ . The ray is then reflected from this point along the line, $y+2 x=1$ . Then the equation of the line of incidence of the ray of light is:

(A) $41 x-38 y+38=0$

(B) $41 x+25 y-25=0$

(D) $41 x-25 y+25=0$

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10. A straight line through origin 0 meets the lines $3 y=10-4 x$ and $8 x+6 y+5=0$ at points $A$ and $B$ respectively. Then 0 divides the segment $A B$ in the ratio:

(A) $2: 3$

(B) $1: 2$

(D) $3: 4$

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11. If a variable line drawn through the intersection of the lines $\frac{x}{3}+\frac{y}{4}=1$ and $\frac{x}{4}+\frac{y}{3}=1$ , meets the coordinate axes at $A$ and $B$ , ( $A \neq B$ ), then the locus of the midpoint of $A B$ is:

(A) $6 x y=7(x+y)$

(B) $4(x+y)^{2}-28(x+y)+49=0$

(D) $14(x+y)^{2}-97(x+y)+168=0$

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12. The point $(2,1)$ is translated parallel to the line $L$ : $x-y=4$ by $2 \sqrt{3}$ units. If the newpoint $Q$ lies in the third quadrant, then the equation of the line passing through $Q$ and perpendicular to $L$ is:

(A) $\mathrm{x}+\mathrm{y}=2-\sqrt{6}$

(B) $x+y=3-3 \sqrt{6}$

(D) $2 x+2 y=1-\sqrt{6}$

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13. Two sides of a rhombus are along the lines, $x-y+1=0$ and $7 x-y-5=0$ . If its diagonals intersect at $(-1,-2)$ , then which one of the following is a vertex of this rhombus?

(A) $\left(\frac{1}{3},-\frac{8}{3}\right)$

(B) $\left(-\frac{10}{3},-\frac{7}{3}\right)$

(D) $(-3,-8)$

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14. The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0,0)(0,41)$ and $(41,0)$ is:

(A) 82

(B) 780

(D) 861

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15. Let $P S$ be the median of the triangle with vertices $P(2,2), Q(6,-1)$ and $R(7,3)$ . The equation of the line passing through $(1,-1)$ and parallel to $PS$ is:

(A) $4 x+7 y+3=0$

(B) $2 x-9 y-11=0$

(D) $2 x+9 y+7=0$

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16. Let $a, b, c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4 a x+2 a y+c=0$ and $5 b x+2 b y+d=0$ lies in the fourth quadrant and is equidistant from the two axes then:

(A) $3 b c-2 a d=0$

(B) $3 b c+2 a d=0$

(D) $2 b c+3 a d=0$

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17. A ray of light along $x+\sqrt{3} y=\sqrt{3}$ gets reflected upon reaching $X$ -axis, the equation of the reflected ray is:

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

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

(D) $\sqrt{3} y=x-1$

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18. The $x$ -coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as $(0,1)(1,1)$ and $(1,0)$ is:

(A) $2+\sqrt{2}$

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

(D) $1-\sqrt{2}$

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19. If the line $2 x+y=k$ passes through the point which divides the line segment joining the points $(1,1)$ and $(2,4)$ in the ratio $3: 2$ , then $k$ equals:

(A) $\frac{29}{5}$

(B) 5

(D) $\frac{11}{5}$

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20. The lines $L_{1}: y-x=0$ and $L_{2}: 2 x+y=0$ intersect the line $L_{3}: y+2=0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_{1}$ and $L_{2}$ intersects $L_{3}$ at $R$ .

Statement-1: The ratio $P R: R Q$ equals $2 \sqrt{2}: \sqrt{5}$
Statement-2: In any triangle, bisector of an angle divide the triangle into two similar triangles.

(A) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B) Statement-1 is true, Statement-2 is false.

(D) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

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