Straight lines practice · Medium

The lines $a x+b y+c=0, \quad b x+c y+a=0$ and $c x+a y+b=0$ are concurrent only when

If $L_1$ is line joining points $\left(\frac{5}{2}, 0\right)$ and $(0,-5),$ and $L_2$ is line joining points $\left(\frac{1}{2}, \frac{1}{2}\right)$ and $(0,1)$ and $L_3: 3 x-y-k=0$ are concurrent lines then $k$ is

A line through $A(-5,-4)$ meets the lines $x+3 y+2=0,2 x+y+4=0$ and $x-y-5=0$ at $B, C$ and $D$ respectively. If $\left(\frac{15}{A B}\right)^2+\left(\frac{10}{A C}\right)^2=\left(\frac{6}{A D}\right)^2$, then the equation of the line is

Points on the line $y = x$ whose perpendicular distance from the line $3x + 4y = 12$ are 4 have the coordinates

A variable line through the point $(6 / 5,6 / 5)$ cuts the coordinate axes in the points $A$ and $B$. If the point $P$ divides $A B$ internally in the ratio $2: 1$, then the locus of $P$ 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

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 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

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

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 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

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

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 $P\left(\frac{1+t}{\sqrt{2}}, \frac{2+t}{\sqrt{2}}\right)$ is any point on a line, then the range of values of $t$ for which the point $P$ lies between the parallel lines $x+2 y=1$ and $2 x+4 y=15$, is

If $P\left( {\sin \theta ,1/\sqrt 2 } \right)$ and $Q\left( {1/\sqrt 2 ,\cos \theta } \right), - \pi \le \theta \le \pi$ are two points on the same side of the line $x - y = 0,$ then $\theta$ belongs to the interval

A point equidistant from the lines $4\;x + 3\;y + 10 = 0,\;5\;x - 12\;y + 26 = 0$ and $7\;x + 24\;y - 50 = 0$ is