11. Straight Lines(11th)

Straight Lines

1. The line $L$ given by $\frac{x}{5}+\frac{y}{b}=1$ passes through the point $(13,32)$ . The line $K$ is parallel to $L$ and has the equation $\frac{x}{c}+\frac{y}{3}=1$ . Then the distance between $L$ and $K$ is:

(A) $\sqrt{17}$

(B) $\frac{17}{\sqrt{15}}$

(D) $\frac{23}{\sqrt{15}}$

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2. The shortest distance between the line $y-x=1$ and the curve $x=y^{2}$ is:

(A) $\frac{2 \sqrt{3}}{8}$

(B) $\frac{3 \sqrt{2}}{5}$

(D) $\frac{3 \sqrt{2}}{8}$

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3. The lines $p\left(p^{2}+1\right) x-y+q=0$ and $\left(p^{2}+1\right)^{2} x+\left(p^{2}+1\right) y+2 q=0$ are perpendicular to a common line for:

(A) exactly one values of $p$

(B) exactly two values of $p$

(D) no value of $p$

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4. The perpendicular bisector of the line segment joining $P(1,4)$ and $Q(k, 3)$ has $y$ -intercept -4. Then a possible value of $k$ is:

(A) 1

(B) 2

(D) -4

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5. Let $\mathrm{A}(h, k), \mathrm{B}(1,1)$ and $\mathrm{C}(2,1)$ be the vertices of a right angled triangle with $AC$ as its hypotenuse. If the area of the triangle is 1 square unit, then the set of values which ‘ $k$ ‘ can take is given by:

(A) $\{-1,3\}$

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

(D) $\{0,2\}$

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6. Let $P=(-1,0), Q=(0,0)$ and $R=(3,3 \sqrt{3})$ be three point. The equation of the bisector of the angle $P Q R$ is:

(A) $\frac{\sqrt{3}}{2} x+y=0$

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

(D) $x+\frac{\sqrt{3}}{2} y=0$

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7. If one of the lines of $m y^{2}+\left(1-m^{2}\right) x y-m x^{2}=0$ is a bisector of angle between the lines $x y=0$ , then $m$ is:

(A) 1

(B) 2

(D) -2

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8. A straight line through the point $A(3,4)$ is such that its intercept between the axes is bisected at $A$ . Its equation is:

(A) $x+y=7$

(B) $3 x-4 y+7=0$

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

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9. If $\left(a, a^{2}\right)$ falls inside the angle made by the lines $y=\frac{x}{2}, x>0$ and $y=3 x, x>0$ , then $a$ belong to:

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

(B) $(3, \infty)$

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

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10. The line parallel to the $x$ -axis and passing through the intersection of the lines $a x+2 b y+3 b=0$ and $b x-2 a y-3 a=0$ , where $(a, b) \neq(0,0)$ is:

(A) below the $x$ -axis at a distance of $\frac{3}{2}$ from it

(B) below the $x$ -axis at a distance of $\frac{2}{3}$ from it

(D) above the $x$ -axis at a distance of $\frac{2}{3}$ from it

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11. If a vertex of a triangle is $(1,1)$ and the mid points of two sides through this vertex are $(-1,2)$ and $(3,2)$ then the centroid of the triangle is:

(A) $\left(-1, \frac{7}{3}\right)$

(B) $\left(\frac{-1}{3}, \frac{7}{3}\right)$

(D) $\left(\frac{1}{3}, \frac{7}{3}\right)$

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12. If non zero numbers $a, b, c$ are in $H.P.$ , then the straight line $\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0$ always passes through a fixed point. That point is:

(A) $(-1,2)$

(B) $(-1,-2)$

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

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13. The equation of the straight line passing through the point $(4,3)$ and making intercepts on the co-ordinate axes whose sum is -1 is:

(A) $\frac{x}{2}-\frac{y}{3}=1$ and $\frac{x}{-2}+\frac{y}{1}=1$

(B) $\frac{x}{2}-\frac{y}{3}=-1$ and $\frac{x}{-2}+\frac{y}{1}=-1$

(D) $\frac{x}{2}+\frac{y}{3}=-1$ and $\frac{x}{-2}+\frac{y}{1}=-1$

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14. Let $A(2,-3)$ and $B(-2,1)$ be vertices of a triangle $A B C$ . If the centroid of this triangle moves on the line $2 x+3 y=1$ , then the locus of the vertex $C$ is the line:

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

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

(D) $2 x+3 y=9$

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15. If the sum of the slopes of the lines given by $x^{2}-2 c x y-7 y^{2}=0$ is four times their product $c$ has the value:

(A) -2

(B) -1

(D)

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16. If one of the lines given by $6 x^{2}-x y+4 c y^{2}=0$ is $3 x+4 y=0$ , then $c$ equals:

(A) -3

(B) -1

(D) 1

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17. A square of side $a$ lies above the $x$ -axis and has one vertex at the origin. The side passing through the origin makes an angle $\alpha\left(0<\alpha<\frac{\pi}{4}\right)$ with the positive direction of $x$ -axis. The equation of its diagonal not passing through the origin is:

(A) $y(\cos \alpha+\sin \alpha)+x(\cos \alpha-\sin \alpha)=a$

(B) $y(\cos \alpha-\sin \alpha)-x(\sin \alpha-\cos \alpha)=a$

(D) $y(\cos \alpha+\sin \alpha)+x(\sin \alpha+\cos \alpha)=a$

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18. If the pair of straight lines $x^{2}-2 p x y-y^{2}=0$ and $x^{2}-2 q x y-y^{2}=0$ be such that each pair bisects the angle between the other pair, then:

(A) $p q=-1$

(B) $p=q$

(D) $p q=1$

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19. Locus of centroid of the triangle whose vertices are $(a \cos t, a \sin t),(b \sin t,-b \cos t)$ and $(1,0)$ , where $t$ is a parameter, is:

(A) $(3 x+1)^{2}+(3 y)^{2}=a^{2}-b^{2}$

(B) $(3 x-1)^{2}+(3 y)^{2}=a^{2}-b^{2}$

(D) $(3 x+1)^{2}+(3 y)^{2}=a^{2}+b^{2}$

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20. If $x_{1}, x_{2}, x_{3}$ and $y_{1}, y_{2}, y_{3}$ are both in G.P. with the same common ratio, then the points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ and $\left(x_{3}, y_{3}\right)$ :

(A) are vertices of a triangle

(B) lie on a straight line

(D) lie on a circle

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