Straight Lines and Pair of Straight Lines

If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5: 4$, is $a x^2+b y^2+c x y+d x+e y+170=0$, then the value of $a^2+2 b+3 c+4 d+e$ is equal to :

If the pair of lines $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ intersect on the $y$-axis then :

The pair of lines represented by $3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$ are perpendicular to each other for :

If the pair of straight lines ${x^2} - 2pxy - {y^2} = 0$ and ${x^2} - 2qxy - {y^2} = 0$ be such that each pair bisects the angle between the other pair, then :

If the sum of the slopes of the lines given by ${x^2} - 2cxy - 7{y^2} = 0$ is four times their product $c$ has the value :

Let the equation of the pair of lines, y = px and y = qx, can be written as (y $-$ px) (y $-$ qx) = 0. Then the equation of the pair of the angle bisectors of the lines x² $-$ 4xy $-$ 5y² = 0 is :

If $\left( {a,{a^2}} \right)$ falls inside the angle made by the lines $y = {x \over 2},$ $x > 0$ and $y = 3x,$ $x > 0,$ then a belong to :

The set of all possible values of $\theta$ in the interval (0, $\pi$) for which the points (1, 2) and (sin $\theta$, cos $\theta$) lie on the same side of the line x + y = 1 is :

Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (–1, –4) in this line is :

The image of the point (3, 5) in the line x $-$ y + 1 = 0, lies on :