Straight lines practice · Mixed difficulty

The circumcentre of the triangle formed by the lines $y = x,\;y = 2x$ and $y = 3x + 4$ is

The number of lines that are parallel to $2\;x + 6\;y + 7 = 0$ and have an intercept of length 10 between the coordinate axes, is

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

If the straight line $ax + by + c = 0$ always passes through (1,-2) then a,b,c are in

The image of the origin with reference to the line $4x + 3y - 25 = 0,\;$ is

The shortest distance between the two lines $L_1: x=k_1$; $y=k_2$ and $L_2: x=k_3 ; y=k_4$ is equal to

The point of concurrence of the lines $ax + by + c = 0$ and a,b,c satisfy the relation $3a + 2b + 4c = 0$ is

If a circle is inscribed in an equilateral triangle of side $a$, then area of the square inscribed in the circle is

The sides of a rhombus $A B C D$ are parallel to the lines, $x-y+2=0$ and $7 x-y+3=0$. If the diagonals of the rhombus intersect at $P(1,2)$ and the vertex $A$ (different from the origin) is on the $y$-axis, then the ordinate of $A$ is

If the lines $3 x-4 y-13=0,8 x-11 y-33=0$ and $2 x-3 y+\lambda=0$ are concurrent, then $\lambda=$

The line $L$ has intercepts $a$ and $b$ on the coordinate axes. When keeping the origin fixed, the coordinate axes are rotated through a fixed angle, then the same line has intercepts $p$ and $q$ on the rotated axes. Then,

A variable line drawn through the point $(1,3)$ meets the $X$-axis at $A$ and $Y$-axis at $B$. If the rectangle $O A P B$ is completed, where $O$ is the origin, then locus of $P$ is

Family of lines $x{\sec ^2}\theta + y{\tan ^2}\theta - 2 = 0$ for different $\theta$ real , is

Let $A\left( {2, - 3} \right)$ and $B\left( { - 2,1} \right)$ be vertices of a triangle $ABC$. If the centroid of this triangle moves on the line $2x + 3y = 1$, then the locus of the vertex $C$ is the line

The line parallel to $x$-axis and passing through the point of intersection of lines $a x+2 b y+3 b=0$ and $b x-2 a y-3 a=0$, where $(a, b) \neq(0,0)$ is

If points $\left(a^2, 0\right),\left(0, b^2\right)$ and $(1,1)$ are collinear, then

The length of perpendicular from the point $\left( {a\cos \mathrm{\alpha },\;a\sin \alpha } \right)$ upon the straight line $y = x\tan \alpha + c,c > 0$ is

If the algebraic sum of the perpendicular distances from the points $(2,0),(0,2)$ and $(1,1)$ to a variable straight line be zero, then the line passes through the point

If the points (1,2) and (3,4) were to be on the same side of the line $3x - 5y + a = 0,$ then

If the coordinates of ${A}$ and ${B}$ be $(1,1)$ and $(5,7)$, then the equation of the perpendicular bisector of the line segment ${AB}$ is-

The number of points on the line $3x + 4y = 5$, which are at a distance of $\mathrm{se}{\mathrm{c}^2}\;\mathrm{\theta } + 2\mathrm{cosse}{\mathrm{c}^2}\mathrm{\;\theta },\mathrm{\;\theta } \in R$, from the point (1, 3) is

The coordinates of the point $P$ on the line $2 x+3 y+1=0$, such that $|P A-P B|$ is maximum, where $A$ is $(2,0)$ and $B$ is $(0,2)$ is

The value of $x$, so that 2 is the slope of the line through $(2,5)$ and $(x, 3)$, is

Point $P(2,4)$ is translated through a distance $3 \sqrt{2}$ units measured parallel to the line $y-x-1=0$, in the direction of decreasing ordinates, to reach at $Q$. If $R$ is the image of $Q$ with respect to the line $y-x-1=0$, then coordinates of $R$ are

If $A\left( {0,0} \right),B\left( {12,0} \right),C\left( {12,2} \right),D\left( {6,7} \right)$ and $E\left( {0,5} \right)$ are the vertices of the pendagon $ABCDE$, then its area in square units, is

The distance of the point (1, 2) from the line $x + y + 5 = 0$ measured along the line parallel to $3x - y = 7$ is equal to

Acute angle between the lines represented by $\left(x^2+y^2\right) \sqrt{3}=4 x y$ is

The measure of the angle between the lines $x=y$ and $y-0$ is

Consider the following statements.

I. Let $A\left(x_1, y_1\right), B\left(x_2, y_2\right)$ and ${C}\left(x_3, y_3\right)$ be the vertices of a triangle then centroid is $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$

II. If the point $P(x, y)$ divides the line joining the points $A\left(x_1, y_1\right)$ and $B\left(x_2, y_2\right)$ in the ratio $m: n$ (internally), then $x=\frac{m x_2+n x_1}{m+n}, y=\frac{m y_2+n y_1}{m+n}$

Choose the correct option.

If $(\sin \theta, \cos \theta), \theta \in[0,2 \pi]$ and $(1,4)$ lie on the same side or on the line $\sqrt{3} x-y+1=0$, then the maximum value of $\sin \theta$ will be ______.