Straight lines practice · Mixed difficulty

If the lines $x+a y+a=0, b x+y+b=0$ and $c x+c y+1=0\,(a, b, c$ being distinct $\neq 1)$ are concurrent, then the value of $\frac{a}{a-1}+\frac{b}{b-1}+\frac{c}{c-1},$ is

The area of the triangle formed by the points (2, 2), (5, 5), (6, 7) is equal to (in square unit)

From a common point on the lines $x y-3 y-4 x+12=0$ mutually perpendicular lines $L_1$ and $L_2$ are drawn such that they intercept triangles of equal area with coordinate axes. If $\theta_1$ and $\theta_2$ are inclination of $L_1$ and $L_2$ with coordinate axes, then $\left|\theta_1+\theta_2\right|$ is equal to

A straight line $L$ with negative slope passes through the point $(8,2)$ and intersects the positive coordinate axes at point $P$ and $Q$. As $L$ varies the absolute minimum value of $O P+O Q$ is (where, $O$ is origin)

The equation of one side of a rectangle is $3x - 4y - 10 = 0$ and the coordinates of two its vertices are (-2,1) and (2,4). Then, the area of the rectangle is

The line $L$ has intercepts $a$ and $b$ on the coordinate axes. The coordinate axes are rotated through a fixed angle, keeping the origin fixed. If $p$ and $q$ are the intercepts of the line $L$ on the new axes, then $\frac{1}{a^2}-\frac{1}{p^2}+\frac{1}{b^2}-\frac{1}{q^2}$ is equal to

The orthocentre of the triangle with vertices $\left( {2,\frac{{\sqrt 3 - 1}}{2}} \right),\left( {\frac{1}{2}, - \frac{1}{2}} \right)$ and $\left( {2, - \frac{1}{2}} \right)$ is

The values of $k$ for which the line $(k-3) x-\left(4-k^2\right) y+k^2-7 k+6=0$ is parallel to the $x$-axis, is

The lines $2\;x + y - 1 = 0,\;ax + 3\;y - 3 = 0$ and $3x + 2\;y - 2 = 0$ are concurrent for

The least value of $|k|$, so that the lines $x=k+m, y=-2$ and $y=m x$ are concurrent, is

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 the sum of distances from a point P on two mutually perpendicular straight lines is 1 unit, then the locus of P is

If the points $\left(a, a^2\right)$ and $(1,2)$ lie in the same angular region between the lines $3 x+4 y-1=0$ and $2 x+y-3=0$, then

If the distance of any point $(x, y)$ from the origin is defined as $d(x, y)=\max \{|x|,|y|\}$, then the locus of the point $(x, y)$, where $d(x, y)=1$, is

The locus of the point of intersection of the limes $x \cos \alpha+y \sin \alpha=a$ and $x \sin \alpha-y \cos \alpha=b$, where $\alpha$ is a variable, is

Point where all lines of family $x\left( {a + 2b} \right) + y\left( {a + 3b} \right) = a + b$ are concurrent is

The point on the axis of $x$, whose perpendicular distance from the straight line $\frac{x}{a} + \frac{y}{b} = 1$ is $a$, are

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

The points (1, 3) and (5, 1) are the opposite vertices of a rectangle. The other two vertices lie on the line $y = 2x + c$, then the value of c will be

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

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 x+y=4 which are equidistant from the lines $\left| x \right| = \left| y \right|,$ are

A ray of light coming from the point $(1,2)$ is reflected at a point $A$ on the $x$-axis and then passes through the point $(5,3)$. The coordinates of the point $A$ are $(x, y)$. Find $5 x+y$.

The equation of the straight line through the intersection of line $2 x+y=1$ and $3 x+2 y=5$ and passing through the origin, is

If $A\left( {3,\;5} \right),\;B\left( { - 5,\; - 4} \right),\;C\left( {7,\;10} \right)$ are the vertices of a parallelogram, taken in the order, then the coordinates of the fourth vertex are

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

If $x \cos \alpha+y \sin \alpha=-\sin \alpha \tan \alpha$ be the equation of a line, then the length of the perpendiculars on the line from the points $\left(a^2, 2 a\right),(a b, a+b)$ and $\left(b^2, 2 b\right)$ are in

Equation of line passing through point of intersection of $x+y+2=0$ and $x-y+4=0$ and having $x$-intercept $=0$

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

$ABC$ is an isosceles triangle if the coordinates of the base are $B\left( {1,\;3} \right)$ and $C\left( { - 2,\;7} \right)$, the coordinates of vertex $A$ can be