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

If $2a+3b+4c=\gamma$ for some $\gamma$ the family of line $ax+by+c=3$ always passes through a fixed point $\left(\alpha ,\beta \right)$, then the value of $\left[\alpha \beta \gamma \right]$ is _____ (where, $\left[·\right]$ is GIF)

The set of lines $x{\tan }^{-1}a+y{\sin }^{-1}\left(\frac{1}{\sqrt{1+a^2}}\right)+2=0$, where $a\in \left(0,1\right)$, is concurrent at

Locus of the image of the point $\left(1,2\right)$ in the line $\left(x-y+2\right)+\lambda \left(2x-3y+1\right)=0$ where $\lambda$ is parameter is

Equation of the straight line which belongs to the system of straight lines $a\left(2x+y-3\right)+b\left(3x+2 y-5\right)=0$ and is farthest from the pint $\left(4,-3\right)$ is

Let a line $L$ pass through the point of intersection of the lines $bx+10y-8=0$ and $2x-3y=0,b\in R-\left\{\frac{4}{3}\right\}$. If the line $L$ also passes through the point $\left(1,1\right)$ and touches the circle $17\left(x^2+y^2\right)=16$, then the eccentricity of the ellipse $\frac{x^2}{5}+\frac{y^2}{b^2}=1$ is

Equation of the line belonging to the system of lines $\left(x+y-2\right)+\lambda \left(x-y\right)=0$ and parallel to the line $4x-y=0$ is

Given the family of lines $a(2x + y + 4) + b (x - 2 y -3)=0$. The member of the family which makes a $∆$ of the least area with coordinate axes in third quadrant is

$P(2,3)$ is a given point and $a(x+2y-3)+b(x+y-2)=0(a,b\in R)$ is family of lines. If $L_1,L_2$ of the family respectively are at maximum and minimum distance from $P$ then which of the following is/are CORRECT?

Write the equation of the family of lines satisfying the given condition.

Parallel to $x-2y-12=0$.

Write the equation of the family of lines satisfying the given condition.

Parallel to $x+3y-12=0$.

Write the equation of the family of lines satisfying the given condition.

Parallel to $x+4y-12=0$.

Find the equation of the line which passes through the point of intersection of $L_1:2x-3y+5=0\text{ and }{L}_2:3x-2y+5=0$, and also passes through the origin.

Find the equation of diagonals of parallelogram whose vertices when taken in order are $\left(-2,-1\right), \left(3,0\right), \left(1,-2\right), \left(-4,-3\right)$.

Find the equation of diagonals of parallelogram whose vertices when taken in order are $\left(-1,0\right), \left(3,1\right), \left(2,2\right), \left(-2,1\right)$.

Find the equation of diagonals of parallelogram whose vertices when taken in order are $\left(-2,-1\right), \left(4,0\right), \left(3,3\right), \left(-3,2\right)$.

Find the equation of diagonals of parallelogram whose vertices when taken in order are $\left(7,3\right), \left(6,1\right), \left(8,2\right), \left(9,4\right)$.

Find the equation of the straight line which passes through the origin $\left(0,0\right)$ and the point of intersection of the lines $x-y-1=0$ and $2x-3y+1=0$ .

Find the equation of the straight line which passes through the origin $\left(0,0\right)$ and the point of intersection of the lines $x+y+4=0$ and $3x-y-8=0$.

Find the equations of the lines passing through the intersection of the lines $4x-3y-1=0$ and $2x-5y+3=0$ and equally inclined to the axes.