Straight Lines and Pair of Straight Lines

If one of the lines of $$m{y^2} + \left( {1 - {m^2}} \right)xy - m{x^2} = 0$$ is a bisector of angle between the lines $$xy = 0,$$ then $$m$$ is :

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 :

The lines $$p\left( {{p^2} + 1} \right)x - y + q = 0$$ and $$\left( {{p^2} + 1} \right){}^2x + \left( {{p^2} + 1} \right)y + 2q$$ $$=0$$ are perpendicular to a common line for :

A ray of light along $$x + \sqrt 3 y = \sqrt 3$$ gets reflected upon reaching $$X$$-axis, the equation of the reflected ray is :

A ray of light is incident along a line which meets another line, 7x − y + 1 = 0, at the point (0, 1). The ray is then reflected from this point along the line, y + 2x = 1. Then the equation of the line of incidence of the ray of light is :

If the straight line, 2x – 3y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, $$\beta$$), then $$\beta$$ equals :

Suppose that the points (h,k), (1,2) and (–3,4) lie on the line L₁ . If a line L₂ passing through the points (h,k) and (4,3) is perpendicular to L₁ , then $$k \over h$$ equals :

If the two lines x + (a – 1) y = 1 and 2x + a²y = 1 (a$$\in$$R – {0, 1}) are perpendicular, then the distance of their point of intersection from the origin is :

The equation of one of the straight lines which passes through the point (1, 3) and makes an angles $${\tan ^{ - 1}}\left( {\sqrt 2 } \right)$$ with the straight line, y + 1 = 3$${\sqrt 2 }$$ x is :

The portion of the line $4 x+5 y=20$ in the first quadrant is trisected by the lines $\mathrm{L}_1$ and $\mathrm{L}_2$ passing through the origin. The tangent of an angle between the lines $\mathrm{L}_1$ and $\mathrm{L}_2$ is :

If the line segment joining the points $$(5,2)$$ and $$(2, a)$$ subtends an angle $$\frac{\pi}{4}$$ at the origin, then the absolute value of the product of all possible values of $a$ is :

Let two straight lines drawn from the origin $$\mathrm{O}$$ intersect the line $$3 x+4 y=12$$ at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ such that $$\triangle \mathrm{OPQ}$$ is an isosceles triangle and $$\angle \mathrm{POQ}=90^{\circ}$$. If $$l=\mathrm{OP}^2+\mathrm{PQ}^2+\mathrm{QO}^2$$, then the greatest integer less than or equal to $$l$$ is :

Let $$P = \left( { - 1,0} \right),\,Q = \left( {0,0} \right)$$ and $$R = \left( {3,3\sqrt 3 } \right)$$ be three point. The equation of the bisector of the angle $$PQR$$ is :

The lines $${L_1}:y - x = 0$$ and $${L_2}:2x + y = 0$$ intersect the line $${L_3}:y + 2 = 0$$ at $$P$$ and $$Q$$ respectively. The bisector of the acute angle between $${L_1}$$ and $${L_2}$$ intersects $${L_3}$$ at $$R$$. Statement-1: The ratio $$PR$$ : $$RQ$$ equals $$2\sqrt 2 :\sqrt 5$$ Statement-2: In any triangle, bisector of an angle divide the triangle into two similar triangles.

Let ABC be a triangle with A($$-$$3, 1) and $$\angle$$ACB = $$\theta$$, 0 < $$\theta$$ < $${\pi \over 2}$$. If the equation of the median through B is 2x + y $$-$$ 3 = 0 and the equation of angle bisector of C is 7x $$-$$ 4y $$-$$ 1 = 0, then tan$$\theta$$ is equal to :

The combined equation of the two lines $$ax+by+c=0$$ and $$a'x+b'y+c'=0$$ can be written as $$(ax+by+c)(a'x+b'y+c')=0$$. The equation of the angle bisectors of the lines represented by the equation $$2x^2+xy-3y^2=0$$ is :

In a $$\triangle A B C$$, suppose $$y=x$$ is the equation of the bisector of the angle $$B$$ and the equation of the side $$A C$$ is $$2 x-y=2$$. If $$2 A B=B C$$ and the points $$A$$ and $$B$$ are respectively $$(4,6)$$ and $$(\alpha, \beta)$$, then $$\alpha+2 \beta$$ is equal to

Let A $$\left( {h,k} \right)$$, B$$\left( {1,1} \right)$$ and C $$(2, 1)$$ be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is $$1$$ square unit, then the set of values which $$'k'$$ can take is given by :

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $$(0, 0)$$ $$(0, 41)$$ and $$(41, 0)$$ is :