11. Straight Lines(11th)

Straight Lines

1. Let $C$ be the centroid of the triangle with vertices $(3,-1),(1,3)$ and $(2,4)$ . Let $P$ be the point of intersection of the lines $x+3 y-1=0$ and $3 x-y+1=0$ . Then the line passing through the points $C$ and $P$ also passes through the point:

(A) $(-9,-7)$

(B) $(9,7)$

(D) $(-9,-6)$

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2. Let two points be $A(1,-1)$ and $B(0,2)$ . If a point $P\left(x^{\prime}, y^{\prime}\right)$ be such that the area of $\Delta P A B=5$ sq. units and it lies on the line, $3 x+y-4 \lambda=0$ , then a value of $\lambda$ is:

(A) 4

(B) 1

(D) 3

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3. The locus of the mid-points of the perpendiculars drawn from points on the line, $x=2 y$ to the line $x=y$ is:

(A) $3 x-2 y=0$

(B) $7 x-5 y=0$

(D) $5 x-7 y=0$

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4. A straight line $L$ at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of $60^{\circ}$ with the line $x+y=0$ . Then an equation of the line $L$ is:

(A) $x+\sqrt{3} y=8$

(B) $\sqrt{3} x+y=8$

(D) $(\sqrt{3}-1) x+(\sqrt{3}+1) y=8 \sqrt{2}$

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5. Lines are drawn parallel to the line $4 x-3 y+2=0$ , at a distance $\frac{3}{5}$ from the origin. Then which one of the following points lies on any of these lines?

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

(B) $\left(-\frac{1}{4}, \frac{2}{3}\right)$

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

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6. The region represented by $|x-y| \leq 2$ and $|x+y| \leq 2$ is bounded by a:

(A) rhombus of area $8 \sqrt{2}$ sq. units

(B) square of side length $2 \sqrt{2}$ units

(D) rhombus of side length 2 units

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7. If the two lines $x+(a-1) y=1$ and $2 x+a^{2} y=1(a \in R-\{0,1\})$ are perpendicular, then the distance of their point of intersection from the origin is:

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

(B) $\frac{\sqrt{2}}{5}$

(D) $\sqrt{\frac{2}{5}}$

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8. Slope of a line passing through $P(2,3)$ and intersecting the line, $x+y=7$ at a distance of 4 units from $P$ , is:

(A) $\frac{\sqrt{7}-1}{\sqrt{7}+1}$

(B) $\frac{\sqrt{5}-1}{\sqrt{5}+1}$

(D) $\frac{1-\sqrt{7}}{1+\sqrt{7}}$

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9. If the system of linear equations
$x-2 y+k z=1$
$2 x+y+z=2$
$3 x-y-k z=3$
has a solution $(x, y, z), z \neq 0$ , then $(x, y)$ lies on the straight line whose equation is:

(A) $4 x-3 y-4=0$

(B) $3 x-4 y-1=0$

(D) $3 x-4 y-4=0$

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10. Suppose that the points $(h, k),(1,2)$ and $(-3,4)$ lie on the line $L_{1}$ . If a line $L_{2}$ passing through the points $(h, k)$ and $(4,3)$ is perpendicular to $L_{1}$ , then $\frac{k}{h}$ equals:

(A) $\frac{1}{3}$

(B) 3

(D) $-\frac{1}{7}$

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11. A point on the straight line $3 x+5 y=15$ which is equidistant from the coordinate axes will lie only in:

(A) $1^{\text {st }}$ and $2^{\text {nd }}$ qudratants

(B) $4^{\text {th }}$ qudratant

(D) $1^{\text {st }}$ qudratant

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12. Let $O(0,0)$ and $A(0,1)$ be two fixed points. Then the locus of a point $P$ such that the perimeter of $\Delta A O P$ is 4, is:

(A) $9 x^{2}+8 y^{2}-8 y=16$

(B) $8 x^{2}-9 y^{2}+9 y=18$

(D) $9 x^{2}-8 y^{2}+8 y=16$

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13. If a straight line passing through the point $P(-3,4)$ is such that its intercepted portion between the coordinate axes is bisected at $P$ , then its equation is:

(A) $x-y+7=0$

(B) $4 x-3 y+24=0$

(D) $3 x-4 y+25=0$

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14. If the straight line, $2 x-3 y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15, \beta)$ , then $\beta$ equals:

(A) $\frac{35}{3}$

(B) -5

(D) 5

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15. If in a parallelogram $A B D C$ , the coordinates of $A, B$ and $C$ are respectively $(1,2),(3,4)$ and $(2,5)$ , then the equation of the diagonal $A D$ is:

(A) $5 x+3 y-11=0$

(B) $5 x-3 y+1=0$

(D) $3 x+5 y-13=0$

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16. Two sides of a parallelogram are along the lines, $x+y=3 \& x-y+3=0$ . If its diagonals intersect at $(2,4)$ , then one of its vertex is:

(A) $(2,1)$

(B) $(2,6)$

(D) $(3,6)$

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17. Two vertices of a triangle are $(0,2)$ and $(4,3)$ . If its orthocenter is at the origin, then its third vertex lies in which quadrant:

(A) third

(B) fourth

(D) first

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18. If $5,5 r, 5 r^{2}$ are the lengths of the sides of a triangle, then $r$ cannot be equal to:

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

(B) $\frac{5}{4}$

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

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19. A point $P$ moves on the line $2 x-3 y+4=0$ . If $Q(1,4)$ and $R(3,-2)$ are fixed points, then the locus of the centroid of $\Delta P Q R$ is a line:

(A) parallel to $y$ -axis

(B) with slope $\frac{2}{3}$

(D) with slope $\frac{3}{2}$

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20. If the line $3 x+4 y-24=0$ intersects the $x$ -axis at the point $A$ and the $y$ -axis at the point $B$ , then the incentre of the triangle $O A B$ , where 0 is the origin, is:

(A) $(3,4)$

(B) $(2,2)$

(D) $(4,3)$

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