Rectangular Cartesian System Questions with Solutions

12 questions5 easy7 medium

The rectangular Cartesian coordinate system locates any point in a plane by its distances from two perpendicular axes, written as an ordered pair (x, y). It is the starting point for all of coordinate geometry. This topic covers quadrants, the distance formula, the section formula (internal and external division), the midpoint formula, the area of a triangle from its vertices, and collinearity of points. It also covers special points of a triangle such as the centroid, incentre and circumcentre, and the locus of a point. A useful check is that three points are collinear exactly when the area of the triangle they form is zero. Questions here are usually formula-based and can be solved quickly with careful sign handling. Practise the Cartesian coordinate system questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
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
A. 0B. 7C. 2D. 4
mediumAsked in MAH MCA CET 2025View solution ›
Q2.
The projections of a line segment on the $X, Y, Z$ axes are $12, 4, 3$ respectively. The length and direction cosines are
A. $13; \frac{12}{13}, \frac{4}{13}, \frac{3}{13}$ B. $19; \frac{12}{19}, \frac{4}{19}, \frac{3}{19}$ C. $11; \frac{12}{11}, \frac{4}{11}, \frac{3}{11}$ D. $15; \frac{12}{13}, \frac{4}{13}, \frac{3}{13}$
mediumAsked in MAH MCA CET 2025View solution ›
Q3.
The two lines $t y = x + t^{2}$ and $y + t x = 2 t + t^{3}$ intersect at the point lies on the curve whose equation is
A. $y^{2} = 4 x$ B. $y^{2} = - 4 x$ C. $x^{2} = 4 y$ D. $x^{2} = - 4 y$
mediumAsked in AMU MCA 2016View solution ›
Q4.
If $A (cos α, s in α)$ , $B (s in α, - cos α)$ , $C (1, 2)$ are the vertices of a $Δ A B C$ , then as $α$ varies, the locus of its centroid is,
A. $9 (h^{2} + k^{2}) - 6 h - 12 k + 3 = 0$ B. $h^{2} + k^{2} + 6 h - k + 2 = 0$ C. $h^{2} + k^{2} + 7 h + 2 k + 3 = 0$ D. $4 h^{2} + 2 k^{2} - 3 h + 5 k - 12 = 0$
mediumAsked in MAH MCA CET 2024View solution ›
Q5.
The points $(a, b)$ , $(c, d)$ and $(\frac{k c + l a}{k + l}, \frac{k d + l b}{k + l})$ are
A. vertices of an equilateral triangleB. Vertices of an isosceles triangleC. Vertices of a right-angled triangleD. Collinear
mediumAsked in MAH MCA CET 2024View solution ›
Q6.
The point $(4, 1)$ 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 $\frac{π}{4}$ about the origin in the anticlockwise direction The final position of the point is
A. $(\frac{- 1}{2}, \frac{7}{2})$ B. $(\frac{1}{2}, \frac{7}{2})$ C. $(- 2, 72)$ D. $(2, 72)$
mediumAsked in NIMCET 2010View solution ›
Q7.
A line passing through $(4, 2)$ 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
A. $\frac{1}{x} + \frac{1}{y} = 2$ B. $\frac{2}{x} + \frac{1}{y} = 1$ C. $\frac{1}{x} + \frac{2}{y} = 2$ D. $\frac{1}{x} + \frac{1}{y} = \frac{1}{2}$
mediumAsked in NIMCET 2018View solution ›