Rectangular Cartesian System Questions with Solutions

A triangle has a vertex at (1, 2) and the midpoints of the two sides through it are (−1, 1) and (2, 3). Then the area of this triangle is

The perpendicular distance of the point $P(1,2,3)$ from the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$ is

The projections of a line segment on the $X,Y,Z$ axes are $12,4,3$ respectively. The length and direction cosines are

The two lines $ty=x+t^2$ and $y+tx=2t+t^3$ intersect at the point lies on the curve whose equation is

If $A\left(cos\alpha ,sin\alpha \right)$, $B\left(sin\alpha ,-cos\alpha \right)$, $C\left(1,2\right)$ are the vertices of a $\Delta ABC$, then as $\alpha$ varies, the locus of its centroid is,

The points $\left(a,b\right)$, $\left(c,d\right)$ and $\left(\frac{kc+la}{k+l},\frac{kd+lb}{k+l}\right)$ are

If $\left(4,-3\right)$ and $\left(-9,7\right)$ are two vertices of a triangle and $\left(1,4\right)$ is its centroid, then the area of the triangle is

The point $\left(4,1\right)$ undergoes the following transformations successively: (i) Reflection about the line $y=x$ (ii) Translation through a distance of $2$ units along the positive direction of the $x$ -axis (iii) Rotation through an angle $\frac{\pi }{4}$ about the origin in the anticlockwise direction The final position of the point is

If the distance of $\left(x,y\right)$ from the origin is defined as $d\left(x,y\right)=max\left(\left|x\right|,\left|y\right|\right)$, then the locus of points where $d\left(x,y\right)=1$ is:

If $a,b,c$ are the roots of the equation $x^3-3p{x}^2+3qx-1=0$, then the centroid of the triangle with vertices $\left(a,\frac{1}{a}\right)$, $\left(b,\frac{1}{b}\right)$ and $\left(c,\frac{1}{c}\right)$ is the point

If P(1, 2), Q(4, 6), R(5, 7) and S(a, b) are the vertices of a parallelogram PQRS, then

A line passing through $\left(4,2\right)$ meets the x-axis and y-axis at P and Q respectively. If O is the origin, then the locus of the centre of the circumcircle of triangle OPQ is