11. Straight Lines(11th)

Straight Lines

1. If the equation of the locus of a point equidistant from the point $\left(a_{1}, b_{1}\right)$ and $\left(a_{2}, b_{2}\right)$ is $\left(a_{1}-a_{2}\right) x+\left(b_{1}-b_{2}\right) y+c=0$ , then the value of ‘ $c$ ‘ is:

(A) $\sqrt{a_{1}{ }^{2}+b_{1}{ }^{2}-a_{2}{ }^{2}-b_{2}{ }^{2}}$

(B) $\frac{1}{2}\left(a_{2}^{2}+b_{2}^{2}-a_{1}^{2}-b_{1}^{2}\right)$

(D) $\frac{1}{2}\left(a_{1}^{2}+a_{2}^{2}+b_{1}^{2}+b_{2}^{2}\right)$

Click to View Answer

Correct Answer: [Insert Correct Answer]

2. A triangle with vertices $(4,0),(-1,-1),(3,5)$ is:

(A) isosceles and right angled

(B) isosceles but not right angled

(D) neither right angled nor isosceles

Click to View Answer

Correct Answer: [Insert Correct Answer]

3. Locus of mid point of the portion between the axes of $x \cos \alpha+y \sin \alpha=p$ where $p$ is constant is:

(A) $x^{2}+y^{2}=\frac{4}{p^{2}}$

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

(D) $\frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{4}{p^{2}}$

Click to View Answer

Correct Answer: [Insert Correct Answer]

4. If the pair of lines $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$ intersect on the $y$ -axis then:

(A) $2 f g h=b g^{2}+c h^{2}$

(B) $b g^{2} \neq c h^{2}$

(D) none of these

Click to View Answer

Correct Answer: [Insert Correct Answer]

5. The pair of lines represented by $3 a x^{2}+5 x y+\left(a^{2}-2\right) y^{2}=0$ are perpendicular to each other for:

(A) two values of $a$

(B) $\forall a$

(D) for no values of $a$

Click to View Answer

Correct Answer: [Insert Correct Answer]

6. Let a ray of light passing through the point $(3,10)$ reflects on the line $2 x+y=6$ and the reflected ray passes through the point $(7,2)$ . If the equation of the incident ray is $a x+b y+1=0$ , then $a^{2}+b^{2}+3 a b$ is equal to:

Click to View Answer

Correct Answer: [Insert Correct Answer]

7. If the orthocentre of the triangle formed by the lines $2 x+3 y-1=0, x+2 y-1=0$ and $a x+b y-1=0$ , is the centroid of another triangle, whose circumcentre and orthocentre respectively are $(3,4)$ and $(-6,-8)$ , then the value of $|a-b|$ is:

Click to View Answer

Correct Answer: [Insert Correct Answer]

8. Let $A B C$ be an isosceles triangle in which $A$ is at $(-1,0), \angle A=\frac{2 \pi}{3}, A B=A C$ and $B$ is on the positve $x$ -axis. If $\mathrm{BC}=4 \sqrt{3}$ and the line $BC$ intersects the line $y=x+3$ at $(\alpha, \beta)$ , then $\frac{\beta^{4}}{\alpha^{2}}$ is:

Click to View Answer

Correct Answer: [Insert Correct Answer]

9. The lines $\mathrm{L}_{1}, \mathrm{L}_{2}, \ldots, \mathrm{L}_{20}$ are distinct. For $\mathrm{n}=1,2,3, \ldots, 10$ all the lines $\mathrm{L}_{2 \mathrm{n}-1}$ are parallel to each other and all the lines $L_{2 n}$ pass through a given point $P$ . The maximum number of points of intersection of pairs of lines from the set $\left\{\mathrm{L}_{1}, \mathrm{L}_{2}, \ldots, \mathrm{L}_{20}\right\}$ is equal to:

Click to View Answer

Correct Answer: [Insert Correct Answer]

10. Let $A(-2,-1), B(1,0), C(\alpha, \beta)$ and $D(\gamma, \delta)$ be the vertices of a parallelogram $A B C D$ . If the point $C$ lies on $2 x-y=5$ and the point $D$ lies on $3 x-2 y=6$ , then the value of $|\alpha+\beta+\gamma+\delta|$ is equal to:

Click to View Answer

Correct Answer: [Insert Correct Answer]

11. If the sum of squares of all real values of $\alpha$ , for which the lines $2 x-y+3=0, 6 x+3 y+1=0$ and $\alpha x+2 y-2=0$ do not form a triangle is $p$ , then the greatest integer less than or equal to $p$ is:

Click to View Answer

Correct Answer: [Insert Correct Answer]

12. If the line $l_{1}: 3 y-2 x=3$ is the angular bisector of the lines $l_{2}: x-y+1=0$ and $l_{3}: \alpha x+\beta y+17=0$ , then $\alpha^{2}+\beta^{2}-\alpha-\beta$ is equal to:

Click to View Answer

Correct Answer: [Insert Correct Answer]

13. Let the equations of two adjacent sides of a parallelogram $ABCD$ be $2 x-3 y=-23$ and $5 x+4 y=23$ . If the equation of its one diagonal $AC$ is $3 x+7 y=23$ and the distance of $A$ from the other diagonal is $d$ , then $50 \mathrm{~d}^{2}$ is equal to:

Click to View Answer

Correct Answer: [Insert Correct Answer]

14. The equations of the sides $\mathrm{AB}, \mathrm{BC}$ and $CA$ of a triangle $ABC$ are: $2 x+y=0, x+p y=21 a,(a \pm 0)$ and $x-y=3$ respectively. Let $P(2, a)$ be the centroid of $\triangle A B C$ . Then $(B C)^{2}$ is equal to:

Click to View Answer

Correct Answer: [Insert Correct Answer]

15. The equations of the sides $\mathrm{AB}, \mathrm{BC}$ and $CA$ of a triangle $ABC$ are $2 x+y=0, x+\mathrm{p} y=15 \mathrm{a}$ and $x-y=3$ respectively. If its orthocentre is $(2, a),-\frac{1}{2}<\mathrm{a}<2$ , then $p$ is equal to:

Click to View Answer

Correct Answer: [Insert Correct Answer]

16. A ray of light passing through the point $P(2,3)$ reflects on the $x$ -axis at point $A$ and the reflected ray passes through the point $Q(5,4)$ . Let $R$ be the point that divides the line segment $AQ$ internally into the ratio $2: 1$ . Let the co-ordinates of the foot of the perpendicular $M$ from $R$ on the bisector of the angle $PAQ$ be $(\alpha, \beta)$ . Then, the value of $7 \alpha+3 \beta$ is equal to:

Click to View Answer

Correct Answer: [Insert Correct Answer]

17. Let $A\left(\frac{3}{\sqrt{a}}, \sqrt{a}\right), a>0$ , be a fixed point in the $xy$ -plane. The image of $A$ in $y$ -axis be $B$ and the image of $B$ in $x$ -axis be $C$ . If $D(3 \cos \theta, a \sin \theta)$ is a point in the fourth quadrant such that the maximum area of $\Delta \mathrm{ACD}$ is 12 square units, then $a$ is equal to:

Click to View Answer

Correct Answer: [Insert Correct Answer]

18. Let the points of intersections of the lines $x-y+1=0, x-2 y+3=0$ and $2 x-5 y+11=0$ are the mid points of the sides of a triangle $\triangle \mathrm{ABC}$ . Then, the area of the $\triangle \mathrm{ABC}$ is:

Click to View Answer

Correct Answer: [Insert Correct Answer]

19. A man starts walking from the point $P(-3,4)$ , touches the $x$ -axis at $R$ , and then turns to reach at the point $Q(0,2)$ . The man is walking at a constant speed. If the man reaches the point $Q$ in the minimum time, then $50\left((P R)^{2}+(R Q)^{2}\right)$ is equal to:

Click to View Answer

Correct Answer: [Insert Correct Answer]

20. Consider a triangle having vertices $A(-2,3), B(1,9)$ and $C(3,8)$ . If a line $L$ passing through the circum-centre of triangle $A B C$ , bisects line $BC$ , and intersects $y$ -axis at point $\left(0, \frac{\alpha}{2}\right)$ , then the value of real number $\alpha$ is:

Click to View Answer

Correct Answer: [Insert Correct Answer]

21. A square $A B C D$ has all its vertices on the curve $x^{2} y^{2}=1$ . The midpoints of its sides also lie on the same curve. Then, the square of area of $A B C D$ is:

Click to View Answer

Correct Answer: [Insert Correct Answer]

22. Let $\tan \alpha, \tan \beta$ and $\tan \gamma ; \alpha, \beta, \gamma \neq \frac{(2 n-1) \pi}{2}, \mathrm{n} \in \mathrm{N}$ be the slopes of three line segments $\mathrm{OA}, \mathrm{OB}$ and $OC$ , respectively, where $O$ is origin. If circumcentre of $\triangle \mathrm{ABC}$ coincides with origin and its orthocentre lies on $y$ -axis, then the value of $\left(\frac{\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma}{\cos \alpha \cos \beta \cos \gamma}\right)^{2}$ is equal to:

Click to View Answer

Correct Answer: [Insert Correct Answer]

23. The maximum value of $z$ in the following equation $z=6 x y+y^{2}$ , where $3 x+4 y \leq 100$ and $4 x+3 y \leq 75$ for $x \geq 0$ and $y \geq 0$ is:

Click to View Answer

Correct Answer: [Insert Correct Answer]

24. If the line, $2 x-y+3=0$ is at a distance $\frac{1}{\sqrt{5}}$ and $\frac{2}{\sqrt{5}}$ from the lines $4 x-2 y+\alpha=0$ and $6 \mathrm{x}-3 \mathrm{y}+\beta=0$ , respectively, then the sum of all possible values of $\alpha$ and $\beta$ is:

Click to View Answer

Correct Answer: [Insert Correct Answer]

25. Let $A(1,0), B(6,2)$ and $C\left(\frac{3}{2}, 6\right)$ be the vertices of a triangle $A B C$ . If $P$ is a Point inside the triangle $A B C$ such that the triangles $A P C, A P B$ and $B P C$ have equal areas, then the length of the line segment $P Q$ , where $Q$ is the point $\left(-\frac{7}{6},-\frac{1}{3}\right)$ , is:

Click to View Answer

Correct Answer: [Insert Correct Answer]