Cartesian Coordinate System

The area of a triangle is $5$ and two of its vertices are $A\left(2,1\right)$ and $B\left(3,-2\right)$. The third vertex which lies on line $y=x+3$ is

Let the algebraic sum of the perpendicular distances from the points $A\left(2,0\right),B\left(0,2\right),C\left(1,1\right)$to a variable line be zero. Then all such lines

One side of a square of length $a$ is inclined to the $x$-axis at an angle $\alpha$ with one of the vertices of the square at the origin. The equation of a diagonal of the square is

If the equations of the hypotenuse and a side of a right-angled isosceles triangle be $x+my=1$ and $x=k$ respectively, then

Line $\mathrm{L}$ is perpendicular to the line $5x-y=1$. The area of triangle formed by the line and coordinate axes is $5$. Its equation is

If the coordinates of the vertices of a triangle are rational numbers, then which of the following points of the triangle will always have rational coordinates?

One diagonal of a square is the portion of the line $\sqrt{3}x+y=2\sqrt{3}$ intercepted by the axes. Then an extremity of the other diagonal is

If $bx+cy=a,$ where $a, b, c$ are of the same sign, be a line such that the area enclosed by the line and the axes of reference is $\frac{1}{8}$ square units, then 

Let $\mathrm{L}$ be the line $2x+y=2$. If the axes are rotated by $45°$, then the intercepts made by the line $\mathrm{L}$ on the new axes are respectively

A line passing through the point $\left(2,2\right)$ and the axes enclose an area $k$. The intercepts on the axes made by the line are given by the two roots of

$A\left(1,2\right)$ and $B\left(7,10\right)$ are two points. If $P\left(x,y\right)$ is a point such that the angle $APB$ is $60°$ and the area of the triangle $APB$ is maximum, then which of the following is (are) true?

A ray travelling along the line $3x-4y=5$ after being reflected from a line $\mathrm{L}$ travels along the line $5x+12y=13$. Then the equation of the line $\mathrm{L}$ is

A ray of light travelling along the line $x+y=1$ is incident on the $x$-axis and after refraction it enters the other side of the $x$-axis by turning $30°$ away from the $x$-axis. The equation of the line along which the refracted ray travels is

All points lying inside the triangle formed by the points $\left(1,3\right),\left(5,0\right)$ and $\left(-1,2\right)$ satisfy

If ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\beta }_1,{\beta }_2,{\beta }_3$ are the values of $n$ for which $\sum _{r=0}^{n-1}{x}^{2r}$ is divisible by $\sum _{r=0}^{n-1}{x}^r$, then the triangle having vertices $\left({\alpha }_1,{\beta }_1\right),\left({\alpha }_2,{\beta }_2\right)$ and $\left({\alpha }_3,{\beta }_3\right)$ cannot be

Two equal sides of an isosceles triangle are the lines $\left(a+b\right)x+\left(a-b\right)y-2 ab=0$ and $\left(a-b\right)x+\left(a+b\right)y-2ab=0$. If the third side passes through the point $\left(b-a, a-b\right)$, then the equation of third side may be

The equation of two equal sides $AB$ and $AC$ of an isosceles triangle $ABC$ are $x+y=5$ and $7x-y=3$ respectively. If area of $∆ABC$ is $5$ square units, then equation of $BC$ may be

The integral values of $\beta$ so that point $\left(0,\beta \right)$ lies on or inside the triangle having sides $y+3x+2=0, 3y-2x-5=0$ and $4y+x-14=0$ may be

If $\left(\alpha ,\beta \right)$ is the foot of perpendicular from $\left(x_1,y_1\right)$ to line $lx+my+n=0$, then

Number of points $\left(x,y\right)$ having integral coordinates satisfying the condition $x^2+y^2<25$ is