Pair of Straight Lines

Show that the product of the perpendicular distances from a point $\left(\alpha ,\beta \right)$ to the pair of straight lines $a{x}^2+2hxy+b{y}^2=0$ is $\frac{\left|a{\alpha }^2+2h\alpha \beta +b{\beta }^2\right|}{\sqrt{{\left(a-b\right)}^2+4h^2}}$.

If $\theta$ is the angle between the pair of lines $a{x}^2+2hxy+b{y}^2=0$ then prove that $\cos \left(\theta \right)=\frac{\left|a+b\right|}{\sqrt{{\left(a-b\right)}^2+4h^2}}$.

The combined equation of the pair of line passing through $(-1,2)$, one is parallel to $x+3y-1=0$ and other is perpendicular to $2x-3y-1=0$ is

Homogenize $y^2=x$ with $x+y=4$.

Homogenize $y^2=x$ with $x+y=3$.

Homogenize $y^2=x$ with $x+y=2$.

Find the angle between the lines joining the origin to the points of intersection of $y^2=x$ and $x+y=1$.

If the lines $2x^2+(1+k)xy-5y^2=0$ are equally inclined to co-ordinate axes, then find the value of $k$.

If the lines $x^2+(1-k)xy-3y^2=0$ are equally inclined to co-ordinate axes, then find the value of $k$.

If the lines $x^2+(2+k)xy-4y^2=0$ are equally inclined to co-ordinate axes, then find the value of $k$.

If the lines $x^2+2xy-35y^2-4x+44y-12=0$ and $5x+\lambda y-8=0$ are concurrent then $\lambda$ equals

If a pair of lines drawn through the origin forms an isosceles right-angled triangle with the line $2x+3y=6$, then those lines are

If the lines $x^2+kxy+y^2=0$ and $x+y=1$ make up the sides of an equilateral triangle, then the value of $k^2$ equals

The distance between pair of parallel lines represented by $x^2+2xy+y^2-8ax-8ay-9a^2=0$ is _____ units

If $h^2=ab,$ then the slopes of lines represented by $a{x}^2-2hxy+b{y}^2=0$ would be in the ratio $\_\_\_\_\_\_\_\_\_$

The value of $\lambda$, for which the equation $x^2-y^2-x+\lambda y-2=0$represents a pair of straight lines, are

If slopes of lines represented by $K{x}^2+5xy+y^2=0$ differ by $\text{1}$ then $K=$

The lines represented by the equation $x^2+2\sqrt{3}xy+3y^2-3x-3\sqrt{3}y-4=0$, are

If $\theta$ is the angle between the pair of straight lines $x^2-5xy+4y^2+3x-4=0$$,$ then ${\tan }^2\theta$ is equal to

If $x^2+y^2+2gx+2fy+1=0$ represents a pair of straight lines, then $f^2+g^2$ is equal to