2d geometry system practice · Mixed difficulty

If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5: 4$, is $a x^2+b y^2+c x y+d x+e y+170=0$, then the value of $a^2+2 b+3 c+4 d+e$ is equal to:

If the vertices $A, B$ and $C$ of a triangle are $(2,1),(5,2)$ and $(3,4)$, respectively. Then, the circumcentre is

In a parallelogram $A B C D,$ $A(1,2),$ $B(3,4),$ $C(2,5)$ then length of the side $A D$ is equal to

If each of the vertices of a triangle has integral coordinates, then the triangle will not be

The area of the triangle formed by the lines $y=m_{1} x+c_{1}$, $y=m_{2} x+c_{2}$ and $x=0$ is

Let $B$ and $C$ be the two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y-2 x=2$ such that $\triangle A B C$ is an equilateral triangle. Then, the area of the $\triangle A B C$ is

If the points $O(0,0), A(\cos \alpha, \sin \alpha)$, $B(\cos \beta, \sin \beta)$ are the vertices of a right-angled triangle, then $\left|\sin \left(\frac{\alpha-\beta}{2}\right)\right|=$

If $O$ is the origin and the coordinates of $A$ and $B$ are $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ respectively then $O A \times O B \cos \angle A O B$ is equal to

If the points $(a, b),\left(a^{\prime}, b^{\prime}\right)$ and $\left(a-a^{\prime}, b-b^{\prime}\right)$ are collinear, then

The area of the parallelogram formed by the lines $4 y-3 x=1,$ $4 y-3 x-3=0,$ $3 y-4 x+1=0,$ $3 y-4 x+2=0$ is

For what value of '$a$' the lines $x=3, y=4$ and $4 x-3 y+a=0$ are concurrent

If two vertices of an equilateral triangle have integral coordinates, then the third vertex will have

Three points $(p+1,1),(2 p+1,3)$ and $(2 p+2,2 p)$ are collinear, if $p=$

If $(-6,-4),(3,5),(-2,1)$ are the vertices of a parallelogram, then remaining vertex cannot be

The circumcentre of the triangle formed by the lines $x+y=0, x y=0$ and $x \,\,\,7=0$ is

If points $O(0,0), A(3, \sqrt{3})$ and $B(3, a)$ are the vertices of an equilateral triangle, then $a=$

The distance between the points $P\left(2,60^{\circ}\right)$ and $Q\left(5,270^{\circ}\right)$ is

Incentre of triangle, whose vertices are $A(-36,7), B(20,7)$ and $C(0,-8)$, is

The co-ordinates of the foot of perpendicular from the point $(2,3)$ on the line $x+y-11=0$ are

If $\mathrm{O}^{\prime}$ is the orthocentre of triangle $A B C$ whose vertices are at $A\left(a t_1{ }^2, 2 a t_1\right), B\left(a t_2{ }^2, 2 a t_2\right)$ and $C\left(a t_3{ }^2, 2 a t_3\right)$, then the coordinates of the orthocentre of $\triangle O^{\prime} B C$ are

The points $(0,0),(a, 0)$ and $\left(\frac{a}{2}, \frac{a \sqrt{3}}{2}\right)$ are vertices of

The equation to the locus of a point which moves so that its distance from $x$-axis is always one half its distance from the origin, is

The incentre of the triangle formed by the lines $\mathrm{xy}=0$ and $\frac{x}{3}+\frac{y}{4}=1$ is $(\mathrm{h}, \mathrm{k})$, then

The coordinates of two points $A$ and $B$ are $(3,4)$ and $(5,-2)$ respectively. If $P$ is a point not lying on any of the coordinate axes such that $P A=P B$ and Area of $\triangle P A B=10$, then the coordinates of $P$, are

If points $(5,5),(10, k)$ and $(-5,1)$ are collinear, then $k=$

The length of a rectangle is three times the breadth. If the minimum perimeter of the rectangle is $160 \mathrm{~cm}$, then what can you say about breadth?

If the point $P\left(x_1+t\left(x_2 x_1\right), y_1+t\left(y_2 y_1\right)\right)$ divides the line segment joining $A\left(x_1, y_1\right)$ and $B\left(x_2, y_2\right)$ internally, then $\ldots .$.

A point $P$ moves so that its distance from the point $(a, 0)$ is always equal to its distance from the line $x+a=0$. The locus of the point is

If the vertices of a triangle are $(5,2),(2 / 3,2)$ and $(-4,3)$, then the area of the triangle is

Let $A\left(\alpha, \frac{1}{\alpha}\right) B\left(\beta, \frac{1}{\beta}\right)$ and $C\left(\gamma, \frac{1}{\gamma}\right)$ be the vertices of a $\triangle A B C$, where $\alpha, \beta$ are the roots of the equation $x^2-6 p_1 x+2=0$ and $\beta, \gamma$ are the roots of the equation $x^2-6 p_2 x+3=0$ and $\gamma, \alpha$ are the roots of the equation $x^2-6 p_3 x+6=0, p_1, p_2, p_3$ being positive. Then, the coordinates of the centroid of $\triangle A B C$ are