2d geometry system practice · Mixed difficulty

Let $A(a, b)$ be a fixed point and $O$ be the origin of coordinates. If $A_1$ is the mid-point of $O A, A_2$ is the mid-point of $A A_1, A_3$ is the mid-point of $A A_2$ and so on. Then the coordinates of $A_n$ are

The $x$-coordinates of the incentre of the triangle where the mid-points of the sides are $(0,1),(1,1)$ and $(1,0)$ is

Let A be the set of all points $\left( {\alpha ,\beta } \right)$ such that the area of triangle formed by the points (5, 6), (3, 2) and $\left( {\alpha ,\beta } \right)$ is 12 sq units. Then, the least possible length of a line segment joining the origin to a point in A, is

If the vertices of a triangle be $\left(a m_{1}^{2}, 2 a m_{1}\right),\left(a m_{2}^{2}, 2 a m_{2}\right)$ and $\left(a m_{3}^{2}, 2 a m_{3}\right)$, then the area of the triangle is

The polar coordinates of a point having cartesian coordinates $(-1,-1)$ are

If area of $\Delta A B C$ 5 sq. unit where $A(1,1), B(2,2)$ and $C$ lies on $y=2 x$ then co-ordinate of $C$ can be

The medians $A D$ and $B E$ of a triangle with vertices at $A(0, b), B(0,0)$ and $C(a, 0)$ are perpendicular to each other if

The locus of the mid-point of the distance between the axes of the variable line $x \cos \alpha+y \sin \alpha=p$, where $p$ is constant, is

If $O$ is the origin and $P\left(x_1, y_1\right), Q\left(x_2, y_2\right)$ are two points $O P \times O Q \sin \angle P O Q=$

Coordinates of the orthocentre of the triangle whose sides are $x=3, y=4$ and $3 x+4 y=6$ is

If the centroid and circumcentre of $a$ triangle are $(3,3)$ and $(6,2)$ respectively, then the orthocentre, is

Two vertices of a triangle are $(5,4)$ and $(-2,4)$. If its centroid is $(5,6)$ then the third vertex has the coordinates

If the points $(k, 2-2 k),(1-k, 2 k)$ and $(-k-4,6-2 k)$ be collinear, then the possible values of $k$ are

A point moves in such a way that its distance from $(1,-2)$ is always the twice from $(-3,5)$, the locus of the point is

If the area of the triangle with vertices $(x, 0),(1,1)$ and $(0,2)$ is 4 square units then a value of $x$ is

If $\alpha_1, \alpha_2, \alpha_3, \beta_1, \beta_2, \beta_3$ are the values of $n$ for which $\sum\limits_{r = 0}^{n - 1} {{x^{2r}}}$ is divisible by $\sum\limits_{r = 0}^{n - 1} {{x^r}} ,$ 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

If equation of three sides of a triangle are $x=2, y+1=0$ and $x+2 y=4$ then co-ordinates of circumcentre of this triangle are

If the points $(a, 0),(0, b)$ and $(1,1)$ are collinear, then

If the line through the points $(h, 7)$ and $(2,3)$ intersects the line $3 x-4 y-5=0$ at right angles, then the value of $h$ is

If $H\left(\frac{13}{2}, \frac{11}{2}\right)$ is the orthocentre and $P\left(\frac{\sqrt{7}}{2}, \frac{\sqrt{5}}{2}\right)$ is the circumcentre of a triangle, then coordinates of the centroid are ...... .

If the vertices of a triangle $A B C$ are $A(-4,-1)$, $B(1,2)$ and $C(4,-3)$, then the coordinates of the circumcentre of the triangle are,

The foot of the coordinates drawn from $(2,4)$ to the line $x+y=1$ is

Area of the parallelogram formed by the lines
$a_1 x+b_1 y+c_1=0, a_1 x+b_1 y+d_1=0$ and
$a_2 x+b_2 y+c_2=0, a_2 x+b_2 y+d_2=0$ is

The area of the triangle formed by the points $(k, 2-2 k),(-k+1,2 k)$ and $(-4-k, 6-2 k)$ is 70 units. For

The coordinates of the point $A$ and $B$ are $(a k, 0)$ and $\left(\frac{a}{k}, 0\right),(k= \pm 1)$. If a point $P$ moves so that $P A=k P B$, then the equation to the locus of $P$ is

The area of the triangle formed by the points $(a, b+c),(b, c+a),(c, a+b)$ is

The sum of squares of the distances of a moving point from two fixed points $(a, 0)$ and $(-a, 0)$ is equal to a constant. Then, the equation of its locus is

Let $k$ be an integer such that the triangle with vertices $(k,-3 k),(5, k)$ and $(-k, 2)$ has area 28 sq units. Then, the orthocentre of this triangle is at the point

If $t_1, t_2$ and $t_3$ are distinct and the points $\left(t_1, 2 a t_1+a t_1^3\right),\left(t_2, 2 a t_2+a t_2^3\right)$,
$\left(t_3, 2 a t_3+a t_3^3\right)$ are collinear, then

The area of the triangle with vertices
$\begin{aligned} & {[(a+1)(a+2),(a+2)][(a+2)(a+3),(a+3)]} \\ & \text { and }[(a+3)(a+4),(a+4)] \text { is } \end{aligned}$