Point Slope Form of a Line

Q.

The angle between the lines $y=(2-\sqrt{3})x+5$ and $y=\left(2+\sqrt{3}\right)x-7$ is

1. $30°$

2. $60°$

3. $45°$

4. $90°$

Q.

Write an equation in slope-intercept form of the line that passes through $\left(6,-1\right)$ and $\left(3,-7\right)$.

Q. The equation of line whose slope is $\frac{1}{3}$ and $x-$ intercept is $4$ will be

1. $x-3y+12=0$
2. $x-6y-2=0$
3. $4x-y-16=0$
4. $x-3y-4=0$

Q.

Find the equation of a line that is perpendicular to line $g$ that contains $\left(\text{P, Q}\right)$. Coordinate plane with line $g$ that passes through the points $(-2,6)$and $(-3,2)$.

1. $4x+y=Q+4P$

2. $x-4y=-4Q+P$

3. $-4x+y=Q-4P$

4. $x+4y=4Q+P$

Q. Which of the following is the equation of a line passing through origin with slope equal to 1?

1. 5x + 4y = 0
2. 3x + 4y = 2
3. x – y = 0
4. x + y = 0

Q. Equation of the line passing through (–1, 1) and perpendicular to the line 2x + 3y + 4 = 0, is

1. 2(y – 1) = 3(x + 1)
2. 3(y –1)= –2(x + 1)
3. y – 1 = 2(x + 1)
4. 3(y – 1) = x + 1

Q. The distance of the point $(1,3)$ from the line $2x-3y+9=0$ measured along a line $x-y+1=0$

1. $2\sqrt{2}$
2. $3\sqrt{2}$
3. $4\sqrt{2}$
4. $4\sqrt{2}$

Q. The equation of the straight line, which passes through the point $(2,4)$ and makes an angle $\theta$ with positive $x$axis where $\cos \theta =-\frac{1}{3}$ is

1. $2\sqrt{2}x-y-2\sqrt{2}+4=0$
2. $2\sqrt{2}x+y-4\sqrt{2}-4=0$
3. $\sqrt{2}x+y-2\sqrt{2}-4=0$
4. $x+2\sqrt{2}y-4=0$

Q. A ray of light along $x+\sqrt{3}y=\sqrt{3}$ gets reflected upon reaching $x$-axis, the equation of the reflected ray is

1. $y=x+\sqrt{3}$
2. $\sqrt{3}y=x-\sqrt{3}$
3. $y=\sqrt{3}x-\sqrt{3}$
4. $\sqrt{3}y=x-1$

Q. The line joining two points $A(2,0),B(2+\sqrt{3},1)$ is rotated about $A$ in anti-clockwise direction through an angle of ${30}^{\circ }$. If the coordinates of new position of $B$ is $(h,k)$, then the value of $h^2+k^2$ is

Q. Let $PS$ be the median of the triangle with vertices $P(2,2),Q(6,-1)\text{and}R(7,3)$. The equation of the line passing through $(1,-1)$ and parallel to $PS$ is

1. $4x-7y-11=0$
2. $2x+9y+7=0$
3. $4x+7y+3=0$
4. $2x-9y-11=0$

Q. The distance of the point $(1,3)$ from the line $2x-3y+9=0$ measured along a line $x-y+1=0$

1. $2\sqrt{2}$
2. $3\sqrt{2}$
3. $4\sqrt{2}$
4. $4\sqrt{2}$

Q. Which of the following is the equation of a line passing through origin with slope equal to 1?

1. 5x + 4y = 0
2. 3x + 4y = 2
3. x – y = 0
4. x + y = 0

Q.

A line through the point A(2, 0), which makes an angle of ${30}^{\circ }$ with the positive direction of x-axis is rotated about A in clockwise direction through an angle ${15}^{\circ }$. The equation of the straight line in the new position is

1. $(2-\sqrt{3})x-y-4+2\sqrt{3}=0$

2. $(2-\sqrt{3})x+y-4+2\sqrt{3}=0$

3. $(2-\sqrt{3})x-y+4+2\sqrt{3}=0$

4. $(2-\sqrt{3})x-9y+4+2\sqrt{3}=0$

Q. A triangle has its two sides along the lines $y=m_{1}x$ and $y=m_{2}x$, where $m_1,m_2$ are the roots of $2x^2-5x+2=0.$ If $(2,2)$ is the orthocentre of the triangle, then the equation of the third side is

1. $2x-2y=9$
2. $2x+2y=9$
3. $x+y=9$
4. $x-y=9$

Q. Find the foot of the perpendicular of (3, 6) on the line x - 2y + 4 = 0

1. (5, 2)
2. (4, 4)
3. (-2, 1)
4. (2, 3)

Q. A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept is
(I.I.T.Sc. 1992)

1. $\frac{1}{3}$
2. $\frac{2}{3}$
3. 1
4. $\frac{4}{3}$

Q. A rectangle $ABCD$ has its side $AB$ parallel to line $y=x$ and vertices $A,B$ and $D$ lie on $y=1,x=2$ and $x=-2$, respectively. Locus of vertex ${}^{\prime }{C}^{\prime }$ is

1. $x=5$
2. $x-y=5$
3. $y=5$
4. $x+y=5$

Q. The straight line $x+2y=1$ meets the coordinate axes at $A$ and $B$. A circle is drawn through $A,B$ and the origin. Then the sum of perpendicular distances from $A$ and $B$ on the tangent to the circle at the origin is :

1. $4\sqrt{5}$
2. $\frac{\sqrt{5}}{4}$
3. $2\sqrt{5}$
4. $\frac{\sqrt{5}}{2}$

Q. A perpendicular is drawn from the point $A(1,8,4)$ to the line joining the points $B(0,-1,3)$ and $C(2,-3,-1)$. The co-ordinates of the foot of the perpendicular is:

1. $(\frac{5}{3},\frac{2}{3},\frac{19}{3})$
2. $(\frac{5}{3},\frac{2}{3},\frac{-19}{3})$
3. $(\frac{5}{3},\frac{-2}{3},\frac{19}{3})$
4. $(\frac{-5}{3},\frac{2}{3},\frac{19}{3})$

Q. The coordinates of two consecutive vertices $A$ and $B$ of a regular hexagon $ABCDEF$ are $(1,0)$ and $(2,0),$ respectively. Then the equation of the diagonal $CE$ is

1. $\sqrt{3}x+y=4$
2. $x+\sqrt{3}y+4=0$
3. $x+\sqrt{3}y=4$
4. $x-\sqrt{3}y=4$