Straight lines practice · Mixed difficulty

Area of square having two sides $y=x+1$ and $y=x+2$ is

The co-ordinates of the foot of perpendicular from the point $(2,3)$ on the line $x+y-11=0$ are $(x, y)$. Find $x+y$

Consider three points $P=(-\sin (\beta-\alpha),-\cos \beta)$, $Q=(\cos (\beta-\alpha), \sin \beta)$ and $R(\cos (\beta-\alpha+\theta), \sin (\beta-\theta))$, where $0 < \theta, \alpha, \beta < \frac{\pi}{4}$. Then,

Consider the following statements.

I. Equation of the line passing through $(0,0)$ with slope ${m}$ is ${y}={mx}$

II. Equation of the ${x}$-axis is ${x}=0$.
Choose the correct option.

Given four lines with equations $x + 2y - 3 = 0,\;2x + 3y - 4 = 0,\;3x + 4y - 5 = 0,\;$ $4x + 5y - 6 = 0$ These lines are

A straight line $L$ through the point $(3,-2)$ is inclined at an angle of $60^{\circ}$ to the line $\sqrt{3} x+y=1$. If L also intersects the $x$-axis, then the equation of $L$, is

If $u=a_1 x+b_1 y+c_1=0, v=a_2 x+b_2 y+c_2=0$ and $\frac{a_1}{a_2}=\frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, then $u+k v=0$ represents

Assertion: The distance between the parallel lines
$3 x-4 y+9=0 \text { and } 6 x-8 y-15=0 \text { is } \frac{33}{10}$

Reason: Distance between the parallel lines ${Ax}+{By}+{C}_1=0$ and ${Ax}+{By}+{C}_2=0$, is given by ${d}=\frac{\left|{C}_1-{C}_2\right|}{\sqrt{{A}^2+{B}^2}}$

The equation of the bisector of the obtuse angle between the lines $3x - 4y + 7 = 0$ and $- 12x - 5y + 2 = 0,$ is

The equation of straight line passing through $(-a, 0)$ and making the triangle with axes of area '$T$' is

The harmonic conjugate of $\left( {4, - 2} \right)$ with respect to $\left( {2, - 4} \right)$ and $\left( {7,1} \right)$ is

The two points on the line $x+y=4$ that lie at $a$ unit distance from the line $4 x+3 y=10$, are

Consider the equation $y-y_1=m\left(x-x_1\right)$. In this equation, if $m$ and $x_1$ are fixed and different lines are drawn for different values of $y_1$, then

Consider the following statements.

I. The distance between two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is given by
$\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$

II. The coordinates of the mid-point of the line segment joining the points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$
is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

Choose the correct option.

Locus of the 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

The family of lines making an angle ${30^ \circ }$ with the line $\sqrt 3 y = x + 1$ is

The locus of a point P which divides the line joining (1,0) and $\left( {2\cos \theta ,\;2\sin \theta } \right)$ internally in the ratio 2∶3 for all $\theta$, is a

The centroid of a triangle lies at the origin and the coordinates of its two vertices are $(-8,0)$ and $(9,11)$, the area of the triangle in sq. units is

If $A\left(\sin \alpha, \frac{1}{\sqrt{2}}\right)$ and $B\left(\frac{1}{\sqrt{2}}, \cos \alpha\right),-\pi \leq \alpha \leq \pi$, are two points on the same side of the line $x-y=0$, then $\alpha$ belongs to the interval

For $a>b>c>0,$ the distance between $(1,1)$ and the point of intersection of the lines $a x+b y+c=0$ and $b x+a y+c=0$ is less than $2 \sqrt{2}.$ Then

The equation of the lines through the points $(2,3)$ and making an intercept of length 2 units between the lines $y+2 x=3$ and $y+2 x=5$, are

The members of the family of lines $\left( {\lambda + \mu } \right)x + \left( {2\lambda + - \mu } \right)y = \lambda + 2\;\mu ,$ where $\lambda \ne 0,\mu \ne 0,$ pass through the point

The lines parallel to the x-axis and passing through the intersection of the lines $ax + 2by + 3b = 0$ and $bx - 2ay - 3a = 0,$ where $\left( {a,\;b} \right) \ne \left( {0,0} \right)$ is

Joint equation of pair of lines through $\left( {3,\; - 2} \right)$ and parallel to ${x^2} - 4xy + 3{y^2} = 0$ is

If the system of following equations in $x \,\&\, y$

$\begin{aligned} & 2 x+3 y+4=0 ; 3 x+5 y+6=0 ; \\ & 2 x^2+6 x y+5 y^2+8 x+12 y+1=t \end{aligned}$

consists of a solution. Then find the numeric value of $t^4$.

The equation of the lines through the point (3, 2) which makes an angle of $45^\circ$ with the line $x - 2y = 3$ , are

If the line y=mx meets the lines $x + 2\;y - 1 = 0$ and $2\;x - y + 3 = 0$ at the same point, then m is equal to

If the straight lines, $a x+a m y+1=0,$ $b x+(m+1) b y+1=0$ and $c x+(m+2) c y+1=0, m \neq 0$ are concurrent, then $a, b, c$ are in

If Y-intercept of a line is 8 more than its X-intercept and it passes through $(5,-3)$, then the equation of the line is

Consider the straight lines

The correct statement is