Straight lines - forms, intersection and angle between two lines — practice questions

Question 1 · difficulty L2 · understanding

A line passing through the point $$\mathrm{A}(9,0)$$ 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. $$\frac{x}{\sqrt{3}+2}+y=9$$
• C. $$\frac{x}{\sqrt{3}-2}+y=9$$
• D. $$\frac{y}{\sqrt{3}-2}+x=9$$

Question 2 · difficulty L2 · understanding

The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is :

• A. 1
• B. 2
• C. 3
• D. 0

Question 3 · difficulty L2 · understanding

The intersection of three lines x $$-$$ y = 0, x + 2y = 3 and 2x + y = 6 is a :

• A. Right angled triangle
• B. Equilateral triangle
• C. None of the above
• D. Isosceles triangle

Question 4 · difficulty L3 · understanding

From the point $(-1,-1)$, two rays are sent making angles of $45^{\circ}$ with the line $x+y=0$. These rays get reflected from the mirror $x+2 y=1$. If the equations of the reflected rays are $\mathrm{a} x+\mathrm{b} y=9$ and $c x+d y=7, a, b, c, d \in \mathbf{Z}$, then the value of $a d+b c$ is $\_\_\_\_$ .

Question 5 · difficulty L3 · understanding

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 _________.

Question 6 · difficulty L3 · understanding

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 $\mathrm{BC}$ intersects the line $y=x+3$ at $(\alpha, \beta)$, then $\frac{\beta^4}{\alpha^2}$ is __________.

Question 7 · difficulty L3 · understanding

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 ___________.

Question 8 · difficulty L3 · understanding

Let the angles made with the positive $x$-axis by two straight lines drawn from the point $\mathrm{P}(2,3)$ and meeting the line $x+y=6$ at a distance $\sqrt{\frac{2}{3}}$ from the point P be $\theta_1$ and $\theta_2$. Then the value of $\left(\theta_1+\theta_2\right)$ is:

• A. $\frac{\pi}{2}$
• B. $\frac{\pi}{3}$
• C. $\frac{\pi}{12}$
• D. $\frac{\pi}{6}$

Question 9 · difficulty L3 · understanding

Let $\mathrm{A}(1,2)$ and $\mathrm{C}(-3,-6)$ be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line $7 x-y=14$. If $\mathrm{B}(\alpha, \beta)$ and $\mathrm{D}(\gamma, \delta)$ are the other two vertices, then $|\alpha+\beta+\gamma+\delta|$ is equal to :

• A. 3
• B. 6
• C. 1
• D. 9

Question 10 · difficulty L3 · understanding

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
• C. -6
• D. 6