Family of Lines

The condition that the diagonals of the parallelogram formed by the lines ax + by + c = 0; ax + by + c' = 0; a'x + b'y + c = 0 & a'x + b'y + c' = 0 are at right angles is

The straight line $(x-3)+k(y+2)=0$ passes through a fixed point for all the values of $k$. The coordinates of the fixed point are ______

If $a+b+c=0,$ then the coordinates of the fixed point through which the straight line $3ax+5by+2c=0$ passes are _________

The ratio in which the straight line $3x+4y=7$ divides the line segment joining the points $(1, 2)$ and $(-2, 1)$ is _____

For all non-zero real values of $k,$ the line $x+ky+(1-k)=0$ passes through a fixed point. Then the co-ordinates of the point are _________

If $4a+5b+10c=\lambda$, then the value of $\lambda$ for which the line $ax+by+c-2=0$ always passes through a fixed point is

The equation of the line passing through the point of intersection of the lines $2x+y-4=0$, $x-3y+5=0$ and lying at a distance of $\sqrt{5}$ units from the origin, is

Let $3x^2+8xy-3y^2=0$ represent the lines $L_1,L_2$ and $3x^2+8xy-3y^2+2x-4y-1=0$ represent the lines $L_3, L_4$. Let $L$ be the line joining the points of intersection of $L_1, L_3$ and $L_2, L_4.$Then, the area (in square units) of the triangle formed by $L$ with the coordinate axes is

If $3a+5b+6c=0$ , then the family of line $ax+by+c=0$ pass through the fixed point _____

A variable line which passes through the point of intersection of the straight lines $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x}{b}+\frac{y}{a}=1$ meets the coordinate axes at points $A$ and $B$ respectively, then find the locus of the midpoint of $AB$:

If distance of any point $(x,y)$ from the origin is defined as $d=$$max\{\left|x\right|,\left|y\right|\},$ then the distance of the common point for the lines given by $x(1+\lambda )+\lambda y=-(2+\lambda )$ (where $\lambda$ is a real parameter) from origin is:

If the algebraic sum of the perpendicular distances from the points $A \left(0, -2\right), B\left(\mathrm{2,} 0\right)$ and $C\left(\mathrm{1,1}\right)$ on a variable  line $ax+by+c=0$ is zero, then the line $ax+by+c=0$ always passes through a fixed point $P$, w.r.t.the$∆BAC$ is its

A line cuts the $x$-axis at $A\left(7,0\right)$ and the $y$-axis at $B\left(0,-5\right)$. A variable line $PQ$ is drawn perpendicular to $AB$ cutting the $x \& y$ axes at $P \& Q$ respectively. If $AQ \& BP$ intersects at $R$, then the locus of $R$ is

If $a^3-2a^{2}b+a{b}^2+a^{2}c-2b^3+b^{2}c=0; a, b, c \in R-\left\{0\right\}$, then line $ax+by+c=0$ are concurrent at the point

A family of lines is given by $(1+2\lambda )x+(1-\lambda )y+\lambda =0,\lambda$ being the parameter. The line belonging to this family at the maximum distance from the point $(1,4)$ is

Family of lines x (a + b) + y = 1 where a and b are the roots of the equation x³ – 3x² + x + 1 = 0 and [a + b] = 1 (where [.] denotes the greatest integer function), such that it intercepts a triangle of area A with co-ordinate axes. Find the maximum value of 2A.

Family of lines x(a + b) + y = 1 where a and b are roots of the equation x³ – 3x² + x + 1 = 0
and [a + b] = 1 (where [.] denotes the greatest integer function) such that it intercepts a triangle of area A with coordinate axes then Amax is ________

Family of lines x(a + b) + y = 1 where a and b are roots of the equation x³ – 3x² + x + 1 = 0
and [a + b] = 1 (where [.] denotes the greatest integer function) such that it intercepts a triangle of area A with coordinate axes then Amax is ________

If$5a + 4b + 20c = t$, then the value of $t$ for which the line $ax + by + c - 1 = 0$ always passes through a fixed point is

If $a^3-8b^3-27c^3=6a^{2}b-12a{b}^2,$ then the family of lines $cx+by+a=0$ are concurrent at