Straight lines practice · Easy

(i) The points $(-1,0),(4,-2)$ and $(\cos 2 \theta, \sin 2 \theta)$ are collinear

(ii) The points $(-1,0),(4,-2)$ and $\left(\frac{1-\tan ^2 \theta}{1+\tan ^2 \theta}, \frac{2 \tan \theta}{1+\tan ^2 \theta}\right)$ are collinear

If the line $2 x+y=k$ passes through the point which divides the line segment joining the points $(1,1)$ and $(2,4)$ in the ratio $3: 2$, then $k$ equals :

Bisector of angle between $x+2 y-1=0$ and $2 x+y+1=0$ containing $(2,1)$ is

The area of the parallelogram formed by the lines $x\cos \alpha + y\sin \alpha = p,\;x\cos \alpha + y\sin \alpha = q,$ $x\cos \beta + y\sin \beta = r$ and $x\cos \beta + y\sin \beta = s$ for given values of $p,q,r$ and s is least, if $\left( {\alpha - \beta } \right) =$

Equation of the line equidistant from the parallel lines $9 x+6 y-7=0$ and $3 x+2 y+6=0$ is

If $a, b, c$ be in A.P., then $a x+b y+c=0$ represents

The distance between the lines $5x - 12y + 65 = 0$ and $5x - 12y - 39 = 0$ is

If $a, b, c \in \mathrm{R}$ and 1 is a root of equation $a x^{2}+b x+c=0$, then the curve $y=4 a x^{2}+3 b x+2 c, a \neq 0$ intersect $x$-axis at

The line segment joining the points $(1,2)$ and $(-2,1)$ is divided by the line $3 x+4 y=7$ in the ratio:

The straight line $3x + y = 9$ divided the line segment joining the points (1, 3) and (2,7) in the ratio

If a variable line drown through the point of intersection of straight lines $\frac{x}{\mathrm{\alpha }} + \frac{y}{\mathrm{\beta }} = 1$ and $\frac{x}{\mathrm{\beta }} + \frac{y}{\mathrm{\alpha }} = 1$ meets the coordinate axes in A and B,then the locus of the mid point of AB is

If $A\left( {1,1} \right),B\left( {\sqrt 3 + 1,\;2} \right)$ and $C\left( {\sqrt 3 ,\sqrt 3 + 2} \right)$ be three vertices of a square, then the diagonal through B is

If orthocenter and circumcentre of a triangle are respectively (1, 1) and (3, 2), then the coordinates of its centroid are

If a straight line through the point $P(3,5)$ makes an angle $\frac{\pi}{6}$ with $x$-axis and meets the lines $2 x+y+5=0 \,\& \,3 x-2 y+7=0$ at $Q \,\&\, R$ then $\frac{P Q}{P R}=$

The distance of point $(-1,1)$ from the line $12 x-5 y+9=0$ is

The incentre of the triangle formed by the lines $x = 0,y = 0$ and $3x\;4y = 12$ is at

The coordinates of foot of the perpendicular drawn from the point $(2,4)$ on the line $x+y=1$, are

The orthocentre of the triangle formed by the lines $x=2, y=3$ and $3 x+2 y=6$ is at the point

The distance between the two points $A$ and $A^{\prime}$ which lie on $y=2$ such that both the line segments $A B$ and $A^{\prime} B$ (where $B$ is the point $(2,3)$ ) subtend angle $\frac{\pi}{4}$ at the origin, is equal to

Two lines are drawn through $(3,4)$, each of which makes angle of $45^{\circ}$ with the line $x-y=2$, then area of the triangle formed by these lines is $N$. Find $2 N$.

If the straight lines $x+3 y=4,3 x+y=4$ and $x+y=0$ form a triangle, then the triangle is

Let $A(1,1), B(6,5), C(3 / 2,2)$ be the vertices of $\triangle A B C$. If a point $P$ is such that areas of $\triangle P A B, \triangle P A C$ and $\triangle P B C$ are equal and $Q\left(-\frac{7}{6},-\frac{1}{3}\right)$ is a point, then $P Q=$

If a point $P\left( {4,\;3} \right)$ is shifted by a distance$\sqrt 2$ unit parallel to the line $y = x$, then coordinates of p in new position are

The equation of the sides of a triangle are $x - 3y = 0$, $4x + 3y = 5$ and $3x + y = 0$. The line $3x - 4y = 0$ passes through

The straight line $x+2 y-9=0,3 x+5 y-5=0$ and $a x+b y-1=0$ are concurrent if the straight line $35 x-22 y+1=0$ passes through the point

If the point (a,a) falls between the lines $\left| {x + y} \right| = 4$, then

The circumcentre of the triangle with vertices (0, 30), (4, 0) and (30, 0) is

The equation of a straight line parallel to $2 x+3 y+11=0$ and which is such that the sum of its intercept on the axes is 15 , is

Consider the following statements about straight lines:

I. Slope of horizontal line is zero and slope of vertical line is undefined.

II. Two lines are parallel if and only if their slopes are equal.

III. Two lines are perpendicular if and only if product of their slope is $-1 .$

Which of the above statements are true?

Let the base of a triangle lie along the line x=a and be of length 2a. The area of this triangle is ${a^2}$ if the vertex lies on the lines