11. Straight Lines(11th)

Straight Lines

1. Let $A$ be a fixed point $(0,6)$ and $B$ be a moving point $(2 t, 0)$ . Let $M$ be the mid-point of $A B$ and the perpendicular bisector of $A B$ meets the $y$ -axis at $C$ . The locus of the mid-point $P$ of $MC$ is:

(A) $3 x^{2}-2 y-6=0$

(B) $3 x^{2}+2 y-6=0$

(D) $2 x^{2}-3 y+9=0$

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2. Let $ABC$ be a triangle with $\mathrm{A}(-3,1)$ and $\angle \mathrm{ACB}=\theta, 0<\theta<\frac{\pi}{2}$ . If the equation of the median through $B$ is $2 x+y-3=0$ and the equation of angle bisector of $C$ is $7 x-4 y-1=0$ , then $\tan \theta$ is equal to:

(A) $\frac{1}{2}$

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

(D) 2

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3. The point $\mathrm{P}(\mathrm{a}, \mathrm{b})$ undergoes the following three transformations successively:
(a) reflection about the line $y=x$ .
(b) translation through 2 units along the positive direction of $x$ -axis.
(c) rotation through angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction.
If the co-ordinates of the final position of the point $P$ are $\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)$ , then the value of $2 a+b$ is equal to:

(A) 13

(B) 9

(D) 7

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4. Two sides of a parallelogram are along the lines $4 x+5 y=0$ and $7 x+2 y=0$ . If the equation of one of the diagonals of the parallelogram is $11 x+7 y=9$ , then other diagonal passes through the point:

(A) $(1,2)$

(B) $(2,2)$

(D) $(1,3)$

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5. Let the equation of the pair of lines, $y=p x$ and $y=q x$ , can be written as $(y-p x)(y-q x)=0$ . Then the equation of the pair of the angle bisectors of the lines $x^{2}-4 x y-5 y^{2}=0$ is:

(A) $x^{2}-3 x y+y^{2}=0$

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

(D) $x^{2}-3 x y-y^{2}=0$

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6. Let the centroid of an equilateral triangle $A B C$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x+y=3$ . If $R$ and $r$ be the radius of circumcircle and incircle respectively of $\Delta A B C$ , then $(R+r)$ is equal to:

(A) $7 \sqrt{2}$

(B) $\frac{9}{\sqrt{2}}$

(D) $3 \sqrt{2}$

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7. The number of integral values of $m$ so that the abscissa of point of intersection of lines $3 x+4 y=9$ and $y=m x+1$ is also an integer, is:

(A) 1

(B) 2

(D) 0

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8. The equation of one of the straight lines which passes through the point $(1,3)$ and makes an angles $\tan ^{-1}(\sqrt{2})$ with the straight line, $y+1=3 \sqrt{2} x$ is:

(A) $4 \sqrt{2} x+5 y-(15+4 \sqrt{2})=0$

(B) $5 \sqrt{2} x+4 y-(15+4 \sqrt{2})=0$

(D) $4 \sqrt{2} x-5 y-(5+4 \sqrt{2})=0$

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9. In a triangle $P Q R$ , the co-ordinates of the points $P$ and $Q$ are $(-2,4)$ and $(4,-2)$ respectively. If the equation of the perpendicular bisector of $P R$ is $2 x-y+2=0$ , then the centre of the circumcircle of the $\triangle P Q R$ is:

(A) $(-1,0)$

(B) $(1,4)$

(D) $(-2,-2)$

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10. Let $A(-1,1), B(3,4)$ and $C(2,0)$ be given three points. A line $y=m x, m>0$ , intersects lines $A C$ and $B C$ at point $P$ and $Q$ respectively. Let $A_{1}$ and $A_{2}$ be the areas of $\triangle A B C$ and $\triangle P Q C$ respectively, such that $A_{1}=3 A_{2}$ , then the value of $m$ is equal to:

(A) 1

(B) 3

(D) $\frac{4}{15}$

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11. The intersection of three lines $x-y=0, x+2 y=3$ and $2 x+y=6$ is a:

(A) Right angled triangle

(B) Equilateral triangle

(D) Isosceles triangle

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12. The image of the point $(3,5)$ in the line $x-y+1=0$ , lies on:

(A) $(x-4)^{2}+(y-4)^{2}=8$

(B) $(x-4)^{2}+(y+2)^{2}=16$

(D) $(x-2)^{2}+(y-4)^{2}=4$

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13. A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$ . Three stones A, B and C are placed at the points $(1,1),(2,2)$ and $(4,4)$ respectively. Then, which of these stones is / are on the path of the man?

(A) A only

(B) All the three

(D) B only

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14. Let $L$ denote the line in the $x y$ -plane with $x$ and $y$ intercepts as 3 and 1 respectively. Then the image of the point $(-1,-4)$ in this line is:

(A) $\left(\frac{11}{5}, \frac{28}{5}\right)$

(B) $\left(\frac{29}{5}, \frac{11}{5}\right)$

(D) $\left(\frac{8}{5}, \frac{29}{5}\right)$

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15. A ray of light coming from the point $(2,2 \sqrt{3})$ is incident at an angle $30^{\circ}$ on the line $x=1$ at the point $A$ . The ray gets reflected on the line $x=1$ and meets $x$ -axis at the point $B$ . Then, the line $A B$ passes through the point:

(A) $(3,-\sqrt{3})$

(B) $(4,-\sqrt{3})$

(D) $\left(3,-\frac{1}{\sqrt{3}}\right)$

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16. If the perpendicular bisector of the line segment joining the points $P(1,4)$ and $Q(k, 3)$ has $y$ -intercept equal to -4, then a value of $k$ is:

(A) $\sqrt{14}$

(B) -4

(D) $\sqrt{15}$

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17. Two vertical poles $A B=15 \mathrm{~m}$ and $C D=10 \mathrm{~m}$ are standing apart on a horizontal ground with points $A$ and $C$ on the ground. If $P$ is the point of intersection of $B C$ and $A D$ , then the height of $P$ (in $m$ ) above the line $A C$ is:

(A) $10 / 3$

(B) 5

(D) 6

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18. If a $\triangle A B C$ has vertices $A(-1,7), B(-7,1)$ and $C(5,-5)$ , then its orthocenter has coordinates:

(A) $(-3,3)$

(B) $(3,-3)$

(D) $\left(-\frac{3}{5}, \frac{3}{5}\right)$

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19. The set of all possible values of $\theta$ in the interval $(0, \pi)$ for which the points $(1,2)$ and $(\sin \theta, \cos \theta)$ lie on the same side of the line $x+y=1$ is:

(A) $\left(0, \frac{\pi}{4}\right)$

(B) $\left(0, \frac{3 \pi}{4}\right)$

(D) $\left(0, \frac{\pi}{2}\right)$

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