Straight Lines

Let a ray of light passing through the point $(3,10)$ reflects on the line $2 x+y=6$ and the reflected ray passes through the point $(7,2)$. If the equation of the incident ray is $a x+b y+1=0$, then $a^2+b^2+3 a b$ is…

The equations of the sides $AB,BC$and $CA$of a triangle $ABC$are $2x+y=0,x+py=15a$and $x-y=3$respectively. If its orthocentre is $\left(2,a\right)$, $-\frac{1}{2}<a<2$, then $p$is equal to

Let the point$P\left(\alpha\,,\beta\,\right)$be at a unit distance from each of the two lines$L_{1}:3x-4y+12=0$, and$L_{2}:8x+6y+11=0$. If$P$lies below$L_{1}$and above$L_{2}$, then$100\left(\alpha\,+\beta\,\right)$is…

Let$\alpha\,,\beta\,,\gamma\,,\delta\,\in\,Z$and let$A\left(\alpha\,,\beta\,\right),B\left(1,0\right),C\left(\gamma\,,\delta\,\right)$and$D\left(1,2\right)$be the vertices of a parallelogram$ABCD$. If$AB=\sqrt{10}$and…

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…

Let $(\alpha\,,\beta\,)$ be the centroid of the triangle formed by the lines $15x-y=82,6x-5y=-4$and $9x+4y=17$. Then $\alpha\,+2\beta\,$ and $2\alpha\,-\beta\,$ are the roots of the equation

A light ray emits from the origin making an angle $30^{\circ}$ with the positive $x$-axis. After getting reflected by the line $x+y=1$, if this ray intersects x-axis at Q, then the abscissa of Q is

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…

Consider a triangle $\mathrm{ABC}$ having the vertices $\mathrm{A}(1,2), \mathrm{B}(\alpha, \beta)$ and $\mathrm{C}(\gamma, \delta)$ and angles $\angle A B C=\frac{\pi}{6}$ and $\angle B A C=\frac{2 \pi}{3}$. If the…

If $\left(\alpha\,,\beta\,\right)$is the orthocenter of the triangle $ABC$ with vertices $A\left(3,-7\right),B\left(-1,2\right)$ and $C\left(4,5\right)$, then $9\alpha\,-6\beta\,+60$ is equal to

Consider the triangles with vertices $A\left(2,1\right),B\left(0,0\right)$and $C\left(t,4\right),t=\left[0,4\right]$. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha\,$ and…

Let $A\left(\alpha\,,-2\right),B\left(\alpha\,,6\right)$ and $C\left(\frac{\alpha\,}{4},-2\right)$ be vertices of a $\Delta\,ABC$. If $\left(5,\frac{\alpha\,}{4}\right)$ is the circumcentre of $\Delta\,ABC$, then which…