Family of Straight 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 it is possible to draw a line that belong to all the given family of lines $y-2x+1+{\lambda }_1\left(2y-x-1\right)=0$, $3y-x-6+{\lambda }_2\left(y-3x+6\right)=0$ and $ax+y-2+{\lambda }_3\left(6x-ay-a\right)=0$, then

If the lines $ax+y+1=0, x+by+1=0$ and $x+y+c=0$ ($a\ne b\ne c$ and none of these is equal to $1$) are concurrent, then show that
$\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1$.

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 $\theta$.

If three points $\left(\frac{a_1}{c_1}, \frac{b_1}{c_1}\right), \left(\frac{a_2}{c_2}, \frac{b_2}{c_2}\right)$ and $\left(\frac{a_3}{c_3}, \frac{b_3}{c_3}\right)$ are collinear, prove that the three straight lines $a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0,a_{3}x+b_{3}y+c_3=0$ are concurrent.

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 $x-3y+21=0$.

If $a+b+c=0$, then show that the three straight lines $ax+by+c=0, bx+cy+a=0, cx+ay+b=0$ are concurrent.

If the lines $ax+a^{2}y+1=0, bx+b^{2}y+1=0, cx+c^{2}y+1=0$ are concurrent, show that at least two of the three constants $a, b, c$ are equal.

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 $\frac{a_{1}x+b_{1}y+c_1}{a_1\alpha +b_1\beta +c_1}=\frac{a_{2}x+b_{2}y+c_2}{a_2\alpha +b_2\beta +c_2}$

If $\frac{x}{a}+\frac{y}{b}=\frac{1}{c}, \frac{x}{b}+\frac{y}{c}=\frac{1}{a}$ and $\frac{x}{c}+\frac{y}{a}=\frac{1}{b}$ are concurrent, then show that $ab+bc+ca=0$

Prove that the straight lines $ax+(b+c)y+d=0, bx+(c+a)y+d=0$ and $cx+(a+b) y+d=0$ are concurrent.

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?

Examine whether the straight lines $x-y+4=0, 2x+3y-6=0$ and $8x+7y-26=0$ are concurrent or not.

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 _____