Straight Lines

Let the centroid of an equilateral triangle $ABC$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x+y=3$. If $R$ and $r$ be the radius of circumcircle and incircle…

Let $A(0,1),B(1,1)$and $C(1,0)$ be the mid-points of the sides of a triangle with incentre at the point $D$. If the focus of the parabola $y^{2}=4ax$ passing through $D$ is $(\alpha\,+\beta\,\sqrt{2},0)$, where…

If the point$\left(\alpha\,,\frac{7\sqrt{3}}{3}\right)$lies on the curve traced by themid-points of the line segments of the lines$x\cos\,\theta\,+y\sin\,\theta\,=7,\theta\,\in\,\left(0,\frac{\pi\,}{2}\right)$between…

The point $P\left(a,b\right)$ undergoes the following threetransformations successively: $\left(a\right)$reflection about the line $y=x.$ $\left(b\right)$translation through $2$units along the positivedirection of…

Let $ABC$ be a triangle with $A(-3,1)$ and $\angle\,ACB=\theta\,,0<\theta\,<\frac{\pi\,}{2}$. 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…

Let $m_{1},m_{2}$ be the slopes of two adjacent sides of a square of side $a$such that $a^{2}+11a+3\left(m_{1}^{2}+m_{2}^{2}\right)=220$. If one vertex of the square is…

Among the statements $(S 1)$ : If $A(5,-1)$ and $B(-2,3)$ are two vertices of a triangle, whose orthocentre is $(0,0)$, then its third vertex is $(-4,-7)$ and (S2) : If positive numbers 2a, b, c are three consecutive…

The intersection of three lines $x-y=0,x+2y=3$ and $2x+y=6$ is a/an

In an isosceles triangle $ABC$, the vertex$A$ is $\left(6,1\right)$and the equation of the base$BC$ is $2x+y=4$. Let the point$B$ lie on the line $x+3y=7$. If $\left(\alpha\,,\beta\,\right)$is the centroid…

A triangle $ABC$ lying in the first quadrant has two vertices as $A\left(1,2\right)$ and $B\left(3,1\right)$. If$\angle\,BAC=90^{o},$and$\operatorname{ar}\,\left(\Delta\,\mathrm{ABC}\right)=5\sqrt{5}$sq. units, then the…

Let$C(\alpha\,,\beta\,)$be the circumcentre of the triangle formed by the lines$4x+3y=69$,$4y-3x=17$, and$x+7y=61$. Then$(\alpha\,-\beta\,)^{2}+\alpha\,+\beta\,$is equal to

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