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

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

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?

Prove that all lines represented by the equation $2(\cos \theta +3 \sin \theta )x+(3 \cos \theta -5 \sin \theta )y-(5 \cos \theta -2 \sin \theta )=0$ pass through a fixed point for all values of $\theta$.

The straight line $lx+my+n=0$ bisects the angle between a pair of straight lines of which one is $px+qy+r=0$. Show that the equation of the other line is $(px+qy+r)\left(l^2+m^2\right)=2(lp+mq)(lx+my+n)$.

Find the equation of the straight line which passes through the point of intersection of the two lines $2x-y+5=0$ and $5x+3y-4=0$ and is perpendicular to the line $x-3y+21=0$.

Show that the equation of the straight line through $(\alpha , \beta )$ and through the point of intersection of the lines $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ is $\frac{a_{1}x+b_{1}y+c_1}{a_1\alpha +b_1\beta +c_1}=\frac{a_{2}x+b_{2}y+c_2}{a_2\alpha +b_2\beta +c_2}$

For what value of $a$, the three straight lines $7x-11y+3=0, 4x+3y-9=0$ and $13x+ay-48=0$ pass through the same point?

Two straight lines, drawn through the origin trisect the portion of the line $6x+5y=30$ intercepted between the axes. Find the equations of the straight lines.

Show that the straight line $(2a+3b)x+(2b-a)y-(4b+a)=0$ always passes through a fixed point; find the co-ordinats of the fixed point.

If $p+q+r=0$ for all positions of the moving straight line $px+qy+r=0$, show that the line always passes through a fixed point; find the coordinates of the fixed point.

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

Find the equation of the line through the intersection of the lines $4x-3y+7=0$ and $2x+3y+5=0$ and the point $\left(-4,5\right)$.