Homogeneous Equation of Second degree

The equation of the line common to the pair of lines $m^2{x}^2-\left(m^2+1\right)xy+y^2=0$ and $m{x}^2-\left(m+1\right)xy+y^2=0$ is:

The condition for the equation $x^2+\left(\lambda +\mu \right)xy+\lambda \mu y^2+x+\mu y=0$ to represent a pair of parallel lines and the distance between them are

Find the centroid and the area of the triangle formed by the following lines $2y^2-xy-6x^2=0, x+y+4=0.$

Find the centroid and the area of the triangle formed by the following lines $2y^2-xy-6x^2=0, x+y+4=0.$

Let the equation $a{x}^2+2hxy+b{y}^2=0$ represents a pair of straight lines. Then the angle $\theta$ between the lines is given by $\cos \theta =\frac{\left|a+b\right|}{\sqrt{{\left(a-b\right)}^2+4h^2}}$

If $a:b:c=1:2:3$ then the lines represented by $a{x}^2+bxy+c{y}^2=0$ are

Let $L_1, L_2$ be the lines represented by the equation $4x^2-5xy+3y^2=0$. Let $L_3, L_4$ be two lines passing through the point $\left(4,3\right)$ such that $L_3$ and $L_4$ are perpendicular to $L_1$ and $L_2$ respectively. If the combined equation of $L_3$ and $L_4$ is $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$, then $af+bg+ch=$

$a{x}^2-4xy-2y^2=0$ represents a pair of lines. If $\theta$ is the angle between these lines, $\cos \theta =\frac{1}{5}$ and the possible values of '$a$' are $a_1$ and $a_2\left(a_1<a_2\right)$ then $a_1+3a_2=$

If a line $L$ is common to the pairs of lines $6x^2-xy-12y^2=0$ and $15x^2+14xy-8y^2=0$, then the combined equation the other two lines is

If the slope of one of the lines represented by $5x^2+\frac{40}{3}xy+k{y}^2=0$ is $3$, then the angle between the pair of lines is

If equation of the line pair through the origin and perpendicular to the line pair $xy-3y^2+y-2x+10=0$ is $\lambda xy+\mu x^2=0$ (where $\lambda \& \mu$ are co-prime) then

If the pair of straight lines $9x^2+axy+4y^2+6x+by-3=0$ represents two parallel lines then

The condition that one of the straight lines given by the equation $a{x}^2+2hxy+b{y}^2=0$ may coincide with one of those given by the equation $a^{\text{'}}{x}^2+2h^{\text{'}}xy+b^{\text{'}}{y}^2=0$, is

Find the joint equations of lines passing through origin and having inclinations $\frac{\mathrm{\pi }}{3} \mathrm{and} \frac{5\mathrm{\pi }}{3}$

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 $\cos \left(\theta \right)=\frac{\left|a+b\right|}{\sqrt{{\left(a-b\right)}^2+4h^2}}$.

Find the degree of the homogeneous equation $f(x,y)=x^3\sin \left(\frac{y}{x}\right)$.

Find the degree of the homogeneous equation $f(x,y)=2x+y$.

Find the degree of the homogeneous equation $f(x,y)=\sqrt{x^8-3x^2{y}^6}$.

Find the degree of the homogeneous equation $f(x,y)=x^3+y^3$.