11. Straight Lines(11th)

Straight Lines

1. If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5: 4$ , is $a x^{2}+b y^{2}+c x y+d x+e y+170=0$ , then the value of $a^{2}+2 b+3 c+4 d+e$ is equal to:

(A) 37

(B) -27

(D) 5

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2. Let a variable line of slope $m>0$ passing through the point $(4,-9)$ intersect the coordinate axes at the points $A$ and $B$ . The minimum value of the sum of the distances of $A$ and $B$ from the origin is:

(A) 3

(B) 15

(D) 25

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3. Let $A(-1,1)$ and $B(2,3)$ be two points and $P$ be a variable point above the line $AB$ such that the area of $\triangle PAB$ is 10. If the locus of $P$ is $ax+by=15$ , then $5a+2b$ is:

(A) $-\frac{12}{5}$

(B) $-\frac{6}{5}$

(D) 4

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4. Let two straight lines drawn from the origin $O$ intersect the line $3x+4y=12$ at the points $P$ and $Q$ such that $\triangle OPQ$ is an isosceles triangle and $\angle POQ=90^{\circ}$ . If $l=OP^{2}+PQ^{2}+QO^{2}$ , then the greatest integer less than or equal to $l$ is:

(A) 42

(B) 46

(D) 44

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5. The vertices of a triangle are $A(-1,3), B(-2,2)$ and $C(3,-1)$ . A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is:

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

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

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

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6. Let $A(a, b), B(3,4)$ and $C(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2 a+3,7 b+5)$ from the line $2x+3y-4=0$ measured parallel to the line $x-2y-1=0$ is:

(A) $\frac{17 \sqrt{5}}{6}$

(B) $\frac{15 \sqrt{5}}{7}$

(D) $\frac{\sqrt{5}}{17}$

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7. Let $\alpha, \beta, \gamma, \delta \in \mathbb{Z}$ and let $A(\alpha, \beta), B(1,0), C(\gamma, \delta)$ and $D(1,2)$ be the vertices of a parallelogram $ABCD$ . If $AB=\sqrt{10}$ and the points $A$ and $C$ lie on the line $3y=2x+1$ , then $2(\alpha+\beta+\gamma+\delta)$ is equal to:

(A) 8

(B) 5

(D) 10

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8. If $x^{2}-y^{2}+2hx xy+2gx+2fy+c=0$ is the locus of a point, which moves such that it is always equidistant from the lines $x+2y+7=0$ and $2x-y+8=0$ , then the value of $g+c+h-f$ equals:

(A) 8

(B) 14

(D) 6

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9. A line passing through the point $A(9,0)$ makes an angle of $30^{\circ}$ with the positive direction of $x$ -axis. If this line is rotated about $A$ through an angle of $15^{\circ}$ in the clockwise direction, then its equation in the new position is:

(A) $\frac{y}{\sqrt{3}+2}+x=9$

(B) $\frac{x}{\sqrt{3}+2}+y=9$

(D) $\frac{y}{\sqrt{3}-2}+x=9$

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10. Let $A$ be the point of intersection of the lines $3x+2y=14, 5x-y=6$ and $B$ be the point of intersection of the lines $4x+3y=8, 6x+y=5$ . The distance of the point $P(5,-2)$ from the line $AB$ is:

(A) $\frac{13}{2}$

(B) 8

(D) 6

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11. The distance of the point $(2,3)$ from the line $2x-3y+28=0$ , measured parallel to the line $\sqrt{3}x-y+1=0$ , is equal to:

(A) $3+4 \sqrt{2}$

(B) $6 \sqrt{3}$

(D) $4 \sqrt{2}$

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12. In a $\triangle ABC$ , suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC$ is $2x-y=2$ . If $2AB=BC$ and the points $A$ and $B$ are respectively $(4,6)$ and $(\alpha, \beta)$ , then $\alpha+2\beta$ is equal to:

(A) 42

(B) 39

(D) 45

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13. Let $R$ be the interior region between the lines $3x-y+1=0$ and $x+2y-5=0$ containing the origin. The set of all values of $a$ , for which the points $(a^{2}, a+1)$ lie in $R$ , is:

(A) $(-3,0) \cup\left(\frac{2}{3}, 1\right)$

(B) $(-3,0) \cup\left(\frac{1}{3}, 1\right)$

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

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14. The portion of the line $4x+5y=20$ in the first quadrant is trisected by the lines $L_{1}$ and $L_{2}$ passing through the origin. The tangent of an angle between the lines $L_{1}$ and $L_{2}$ is:

(A) $\frac{30}{41}$

(B) $\frac{8}{5}$

(D) $\frac{25}{41}$

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15. If $(\alpha, \beta)$ is the orthocenter of the triangle $ABC$ with vertices $A(3,-7), B(-1,2)$ and $C(4,5)$ , then $9\alpha-6\beta+60$ is equal to:

(A) 30

(B) 40

(D) 35

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16. Let $(\alpha, \beta)$ be the centroid of the triangle formed by the lines $15x-y=82, 6x-5y=-4$ and $9x+4y=17$ . Then $\alpha+2\beta$ and $2\alpha-\beta$ are the roots of the equation:

(A) $x^{2}-7x+12=0$

(B) $x^{2}-13x+42=0$

(D) $x^{2}-10x+25=0$

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17. If the point $\left(\alpha, \frac{7 \sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments of the lines $x \cos \theta+y \sin \theta=7, \theta \in\left(0, \frac{\pi}{2}\right)$ between the co-ordinates axes, then $\alpha$ is equal to:

(A) -7

(B) 7

(D) $7 \sqrt{3}$

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18. Let $C(\alpha, \beta)$ be the circumcenter of the triangle formed by the lines $4x+3y=69, 4y-3x=17$ , and $x+7y=61$ . Then $(\alpha-\beta)^{2}+\alpha+\beta$ is equal to:

(A) 15

(B) 17

(D) 18

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19. The straight lines $l_{1}$ and $l_{2}$ pass through the origin and trisect the line segment of the line $L: 9x+5y=45$ between the axes. If $m_{1}$ and $m_{2}$ are the slopes of the lines $l_{1}$ and $l_{2}$ , then the point of intersection of the line $y=\left(m_{1}+m_{2}\right)x$ with $L$ lies on:

(A) $6x-y=15$

(B) $6x+y=10$

(D) $y-2x=5$

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20. The combined equation of the two lines $ax+by+c=0$ and $a^{\prime}x+b^{\prime}y+c^{\prime}=0$ can be written as $(ax+by+c)\left(a^{\prime}x+b^{\prime}y+c^{\prime}\right)=0$ . The equation of the angle bisectors of the lines represented by the equation $2x^{2}+xy-3y^{2}=0$ is:

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

(B) $x^{2}-y^{2}-10xy=0$

(D) $3x^{2}+5xy+2y^{2}=0$

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