2d geometry system practice · Medium

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

The coordinates of the foot of the perpendicular from the point $(2,3)$ on the line $y=3 x+4$ are given by

The locus of $P$ such that area of $\triangle P A B=12$ sq. units, where $A(2,3)$ and $B(-4,5)$ is

The values of $t$, if the area of the pentagon $A B C D E$ is $\frac{45}{2} s q$ units, where $A=(1,3), B=(-2,5), C=(-3,-1), D(0,-2)$ and $E=(2, t)$ and here the points are given in order, are

The orthocentre of the triangle formed by the lines $x+y=1,2 x+3 y=6$ and $4 x-y+4=0$ lies in quadrant

The orthocentre of the triangle formed by the line $2 x^2-x y-6 y^2=0$ and the line $x+y+1=0$, is

If the distance of any point $P$ from the point $A(a+b, a-b)$ and $B(a-b, a+b)$ are equal, then the locus of $P$ is

The ratio in which the line $3 x+y-9=0$ divides the segment joining the points $(1,3)$ and $(2,7)$, is

If $(\alpha, \beta),(\bar{x}, \bar{y})$ and $(u, v)$ are respectively coordinates of the circumoentre, centroid and orthocentre of a triangle. Then,

Two points $A$ and $B$ have coordinates $(1,0)$ and $(-1,0)$ respectively and $Q$ is a point which satisfies the relation $A Q-B Q= \pm 1$. The locus of $Q$ is

Let two points be $A\left( {1, - 1} \right)$ and $B\left( {0,2} \right).$ If a point $P\left( {x',y'} \right)$ be such that the area of $\Delta PAB = 5\,\,\,sq.\,\,units$ and it lies on the line, $3x + y - 4\lambda = 0,$ then a value of $\lambda$ is

The points $A(-4,-1), B(-2,-4), C(4,0)$ and $D(2,3)$ are the vertices of

The area of the triangle formed by the lines $7 x-2 y+10=0,7 x+2 y-10=0$ and $y+2=0$ is

The orthocentre of the triangle with vertices $(-2,-6),(-2,4)$ and $(1,3)$ is

The points $A(2 a, 4 a), B(2 a, 6 a)$ and $C(2 a+\sqrt{3} a, 5 a)($ when $a>0)$ are vertices of

The coordinates of point which divides the line segment joining points $A(0,0)$ and $B(9,12)$ in the ratio $1: 2$, are

If the vertices of triangle are $(0,2),(1,0)$ and $(3,1)$, then the triangle is

If $a, b, c$ and $d$ are the sides of a quadrilateral, then the minimum value of $\frac{a^2+b^2+c^2}{d^2}$ is

The figure formed by four points (1,-2), (3, 6), (5,10) and $(3,2)$ is a

If $A$ is $(2,5), B$ is $(4,-11)$ and $C$ lies on $9 x+7 y+4=0$, then the locus of the centroid of the $\Delta A B C$ is a straight line parallel to the straight line is

The equation of the locus of foot of perpendiculars drawn from the origin to the line passing through a fixed point $(a, b)$, is

A point moves such that its distance from the point $(4,0)$ is half that of its distance from the line $x=16$. The locus of this point is

The distance of any point $P(x, y)$ from the point $Q\left(x_1, y_1\right)$ is given by $d(P, Q)=M a x\{| x-x_1|,| y-y_1 \mid\}$. If $Q(1,2)$ and $d(P, Q)=3$ then locus of $P$ is

The coordinates of incentre of $\triangle A B C$, where $A(-3,-2), B(5,0)$ and $C\left(-\frac{1}{3}, \frac{13}{3}\right)$ are

Let $A\left(x_1, y_1\right), B\left(x_2, y_2\right)$ and $C\left(x_3, y_3\right)$ be the vertices of a triangle such that algebraic sum of perpendicular distance from $A, B$ and $C$ to the variable line $a x+b y+c=0$ is always '0'. If $3 a+2 b+c=0$, then find the value of $\sum\limits_{{i=1}}^3\left(x_i+y_i\right)$.

The locus of a point $P$ which moves in such a way that the segment $O P$, where $O$ is the origin, has slope $\sqrt{3}$ is

If vertices of a quadrilateral are $A(0,0), B(3,4), C(7,7)$ and $D(4,3)$, then quadrilateral $A B C D$ is