Pair of Lines

The circumcentre of a triangle formed by the lines $x y+2 x+2 y+4=0$ and $x+y+2=0$, is

The foci of the conic section $25 x^2+16 y^2-150 x=175$ are

If the equation $12 x^2+7 x y-p y^2-18 x+q y+6=0$ represents a pair of perpendicular straight lines, then

If the angle between two lines represented by $2 x^2+5 x y+3 y^2+7 y+4=0$ is $\tan ^{-1} m$, then $m$ is equal to

The line $\frac{x}{a}-\frac{y}{b}=1$ cuts the $x$-axis at $P$. The equation of the line through $P$ perpendicular to the given line is

To the lines $a x^2+2 h x y+b y^2=0$ the lines $a^2 x^2+2 h(a+b) x y+b^2 y^2=0$, are

If the angles between the pair of straight lines represented by the equation $x^2-3 x y+\lambda y^2+3 x-5 y+2=0$ is $\tan ^{-1} \frac{1}{3}$. Where $\lambda$ is a non-negative real number, then $\lambda$ is

The equation $3 x^2+7 x y+2 y^2+5 x+5 y+2=0$ represents

The combined equation of two lines $L$ and $L_1$ is $2 x^2+a x y+3 y^2=0$ and the combined equation of two lines $L$ and $L_2$ is $2 x^2+b x y-3 y^2=0$. If $L_1$ and $L_2$ are perpendicular, then $a^2+b^2=$

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

Suppose the slopes $m_1$ and $m_2$ of the lines represented by $a x^2+2 h x y+b y^2=0$ satisfy $3\left(m_1-m_2\right)-7=0$ and $m_1 m_2-2=0$. Then, which of the following is true?

The area (in sq. units) of the triangle formed by the straight line $x+y=3$and the angular bisectors of the pair of straight lines $x^{2}-y^{2}+2y=1$ is