11. Straight Lines(11th)

Straight Lines

1. Let the area of the triangle formed by a straight line $L : x + by + c = 0$ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line $L$ makes an angle of $45^\circ$ with the positive $x$ -axis, then the value of $b^2 + c^2$ is:

(A) 90

(B) 83

(D) 97

Click to View Answer

Correct Answer: [Insert Correct Answer]

2. Let the triangle $PQR$ be the image of the triangle with vertices $(1, 3)$ , $(3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$ . If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$ , then $15(\alpha - \beta)$ is equal to:

(A) 21

(B) 19

(D) 24

Click to View Answer

Correct Answer: [Insert Correct Answer]

3. A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x - y + 2 = 0$ and $y + 2 = 0$ , respectively. If the locus of the point $P$ , that divides the rod $AB$ internally in the ratio $2 : 1$ is $9(x^2 + \alpha y^2 + \beta xy + \gamma x + 28y) - 76 = 0$ , then $\alpha - \beta - \gamma$ is equal to:

(A) 24

(B) 22

(D) 23

Click to View Answer

Correct Answer: [Insert Correct Answer]

4. Let the lines $3x - 4y - \alpha = 0$ , $8x - 11y - 33 = 0$ , and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left( \frac{57}{13}, \frac{-40}{13} \right)$ , then $|\alpha\lambda|$ is equal to:

(A) 91

(B) 113

(D) 84

Click to View Answer

Correct Answer: [Insert Correct Answer]

5. Let the points $\left( \frac{11}{2}, \alpha \right)$ lie on or inside the triangle with sides $x + y = 11$ , $x + 2y = 16$ and $2x + 3y = 29$ . Then the product of the smallest and the largest values of $\alpha$ is equal to:

(A) 22

(B) 33

(D) 44

Click to View Answer

Correct Answer: [Insert Correct Answer]

6. If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$ , then the centroid of $\triangle PAB$ lies on the line:

(A) $x + 9y = 36$

(B) $9x - 9y = 32$

(D) $6x - 9y = 20$

Click to View Answer

Correct Answer: [Insert Correct Answer]

7. Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$ . If $m$ is the slope of its third side, then the sum of all possible distinct values of $m$ is:

(A) $-2\sqrt{10}$

(B) 12

(D) 6

Click to View Answer

Correct Answer: [Insert Correct Answer]

8. Let $\triangle ABC$ be a triangle formed by the lines $7x - 6y + 3 = 0$ , $x + 2y - 31 = 0$ and $9x - 2y - 19 = 0$ . Let the point $(h, k)$ be the image of the centroid of $\triangle ABC$ in the line $3x + 6y - 53 = 0$ . Then $h^2 + k^2 + hk$ is equal to:

(A) 47

(B) 37

(D) 36

Click to View Answer

Correct Answer: [Insert Correct Answer]

9. Let the line $x + y = 1$ meet the axes of $x$ and $y$ at $A$ and $B$ , respectively. A right angled triangle $AMN$ is inscribed in the triangle $OAB$ , where $O$ is the origin and the points $M$ and $N$ lie on the lines $OB$ and $AB$ , respectively. If the area of the triangle $AMN$ is $\frac{4}{9}$ of the area of the triangle $OAB$ and $AN : NB = \lambda : 1$ , then the sum of all possible value(s) of $\lambda$ is:

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

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

(D) $\frac{13}{6}$

Click to View Answer

Correct Answer: [Insert Correct Answer]

10. A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $L_1 : 2x + y + 6 = 0$ and $L_2 : 4x + 2y - p = 0$ , $p > 0$ , at the points $A$ and $B$ , respectively. If $AB = \frac{9}{\sqrt{2}}$ and the foot of the perpendicular from the point $A$ on the line $L_2$ is $M$ , then $\frac{AM}{BM}$ is equal to:

(A) 5

(B) 3

(D) 4

Click to View Answer

Correct Answer: [Insert Correct Answer]

11. Consider the lines $x(3\lambda + 1) + y(7\lambda + 2) = 17\lambda + 5$ , $\lambda$ being a parameter, all passing through a point $P$ . One of these lines (say $L$ ) is farthest from the origin. If the distance of $L$ from the point $(3, 6)$ is $d$ , then the value of $d^2$ is:

(A) 10

(B) 20

(D) 30

Click to View Answer

Correct Answer: [Insert Correct Answer]

12. Let the three sides of a triangle are on the lines $4x - 7y + 10 = 0$ , $x + y = 5$ and $7x + 4y = 15$ . Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines $x = 0$ , $y = 0$ and $x + y = 1$ is:

(A) $\sqrt{20}$

(B) 20

(D) 5

Click to View Answer

Correct Answer: [Insert Correct Answer]

13. Let $ABC$ be the triangle such that the equations of lines $AB$ and $AC$ be $3y - x = 2$ and $x + y = 2$ , respectively, and the points $B$ and $C$ lie on $x$ -axis. If $P$ is the orthocentre of the triangle $ABC$ , then the area of the triangle $PBC$ is equal to:

(A) 8

(B) 4

(D) 6

Click to View Answer

Correct Answer: [Insert Correct Answer]

14. If the orthocenter of the triangle formed by the lines $y = x + 1$ , $y = 4x - 8$ and $y = mx + c$ is at $(3, -1)$ , then $m - c$ is:

(A) 0

(B) 2

(D) 4

Click to View Answer

Correct Answer: [Insert Correct Answer]

15. A line passing through the point $P(a, 0)$ makes an acute angle $\alpha$ with the positive $x$ -axis. Let this line be rotated about the point $P$ through an angle $\frac{\alpha}{2}$ in the clockwise direction. If in the new position, the slope of the line is $2 - \sqrt{3}$ and its distance from the origin is $\frac{1}{\sqrt{2}}$ , then the value of $3a^2 \tan^2 \alpha - 2\sqrt{3}$ is:

(A) 8

(B) 4

(D) 6

Click to View Answer

Correct Answer: [Insert Correct Answer]

16. Let $a$ be the length of a side of a square $OABC$ with $O$ being the origin. Its side $OA$ makes an acute angle $\alpha$ with the positive $x$ -axis and the equations of its diagonals are $(\sqrt{3} + 1)x + (\sqrt{3} - 1)y = 0$ and $(\sqrt{3} - 1)x - (\sqrt{3} + 1)y + 8\sqrt{3} = 0$ . Then $a^2$ is equal to:

(A) 48

(B) 16

(D) 32

Click to View Answer

Correct Answer: [Insert Correct Answer]

17. A variable line $L$ passes through the point $(3,5)$ and intersects the positive coordinate axes at the points $A$ and $B$ . The minimum area of the triangle $OAB$ , where $O$ is the origin, is:

(A) 35

(B) 25

(D) 40

Click to View Answer

Correct Answer: [Insert Correct Answer]

18. A ray of light coming from the point $\mathrm{P}(1,2)$ gets reflected from the point $Q$ on the $x$ -axis and then passes through the point $R(4,3)$ . If the point $S(h, k)$ is such that $P Q R S$ is a parallelogram, then $h^{2}$ is equal to:

(A) 60

(B) 70

(D) 90

Click to View Answer

Correct Answer: [Insert Correct Answer]

19. If the line segment joining the points $(5,2)$ and $(2, a)$ subtends an angle $\frac{\pi}{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is:

(A) 4

(B) 8

(D) 2

Click to View Answer

Correct Answer: [Insert Correct Answer]

20. The equations of two sides $AB$ and $AC$ of a triangle $ABC$ are $4 x+y=14$ and $3 x-2 y=5$ , respectively. The point $\left(2,-\frac{4}{3}\right)$ divides the third side $BC$ internally in the ratio $2: 1$ , the equation of the side $BC$ is:

(A) $x+6 y+6=0$

(B) $x-3 y-6=0$

(D) $x-6 y-10=0$

Click to View Answer

Correct Answer: [Insert Correct Answer]