Mathematics · Quantitative Aptitude

Coordinate Geometry and Construction

139 Questions

Coordinate geometry involves finding midpoints, calculating ratios of line segments, and understanding 3D projections. These concepts are crucial for tackling quantitative aptitude sections. Practice these questions to master geometric constructions and coordinate plotting.

Midpoint calculationsSection formulas3D geometry projectionsGeometric construction stepsLine segment ratios

Coordinate Geometry and Construction Questions

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

If the join of the two points $(x _1, y _1)$, $(x _2, y _2)$ is divided by a point R externally in ratio $m : n$ then

  1. x - coordinates is $\dfrac {mx _2 - nx _1}{m - n}$
  2. x - coordinates is $\dfrac {my _2 - ny _1}{m - n}$

  3. Both (a) and (b) above

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

When a point C divides a segment $A(x _1,y _1)$ and $B(x _2,y _2)$ in the ratio $m:n$ externally, we use the section formula to find the coordinates of that point.

The Coordinates of point R will be,

$X=\dfrac{mx _2-nx _1}{m-n}$.

Multiple choice construction : division of a line segment dividing a line segment into three or five equal parts divsion of line segmet in given ratio constructions maths

The line joining points $(3,5)$ and $(2,7)$ is divided by $X-$ axis in the ratio.

  1. $5:7$
  2. $3:2$
  3. $-5:7$
  4. $-3:2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A point on the X-axis has a y-coordinate of 0. Using the section formula, the y-coordinate is (m*y2 + n*y1) / (m+n) = 0. Solving 7m + 5n = 0 gives m/n = -5/7, indicating external division.

Multiple choice construction : division of a line segment dividing a line segment into three or five equal parts divsion of line segmet in given ratio constructions maths

The point $(\dfrac{7}{4},\dfrac{7}{8})$ divides the line segment joining the points (4,-1) and (-2,4) internally in the ratio 3 : 5.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the section formula with points (4, -1) and (-2, 4) and ratio 3:5, the x-coordinate is (3*-2 + 5*4) / 8 = 14/8 = 7/4. The y-coordinate is (3*4 + 5*-1) / 8 = 7/8. The point matches.

Multiple choice construction : division of a line segment dividing a line segment into three or five equal parts divsion of line segmet in given ratio constructions maths

The point which divides the line segment joining $(-2, 4), (2, 7)$ in the ratio $2:1$ externally is

  1. $(6, 10)$
  2. $(2, \dfrac{10}{3})$
  3. $(\dfrac{-4}{3}, \dfrac{2}{3})$
  4. $( \dfrac{2}{3} ,6)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For external division in ratio m:n, the formula is (mx2 - nx1)/(m-n), (my2 - ny1)/(m-n). For (2,1) and points (-2,4), (2,7): x = (2*2 - 1*-2)/(2-1) = 6. y = (2*7 - 1*4)/(2-1) = 10.

Multiple choice construction : division of a line segment dividing a line segment into three or five equal parts divsion of line segmet in given ratio constructions maths

Find the points $A(a, b), B(-a, -b)$ and $P(a^2, ab)$ are collinear then the ratio in which p divides $\overline{AB}$ is 

  1. 1 + a : 1 - a

  2. 1 : a

  3. a : 1

  4. 1 - a : 1 + a
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If P(a^2, ab) lies on AB, then (a^2 - a) / (-a - a^2) = ratio. (a(a-1)) / (-a(1+a)) = -(a-1)/(1+a) = (1-a)/(1+a).

Multiple choice construction : division of a line segment dividing a line segment into three or five equal parts divsion of line segmet in given ratio constructions maths

The ratio in which the line segment joining the points $\left(3,-4\right)$ and $\left(-5,6\right)$ is divided by the $x-$ axis, is

  1. $2:3$
  2. $3:2$
  3. $6:4$
  4. $none\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The x-axis divides a line segment at a point where the y-coordinate is 0. Using the section formula, if the ratio is k:1, the y-coordinate is (k*6 + 1*(-4)) / (k+1) = 0, which gives 6k = 4, or k = 2/3.

Multiple choice construction : division of a line segment dividing a line segment into three or five equal parts divsion of line segmet in given ratio constructions maths

The ratio in which the point $(x _{1} \sin^{2} \theta, y _{1} \cos^{2} \theta)$ divides the line joining $(x _{1}, 0)$ and $(0, y _{1})$ is -

  1. $\tan^{2} \theta : \cot^{2} \theta$
  2. $\cos \theta : \sin \theta $
  3. $\cos^{2} \theta : \sin^{2} \theta$
  4. $(1-\cos \theta) : (1-\sin \theta)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the section formula for a point (x, y) dividing (x1, 0) and (0, y1) in ratio m:n, we get x = n*x1 / (m+n) and y = m*y1 / (m+n). Setting x = x1*sin^2(theta) and y = y1*cos^2(theta) leads to the ratio m:n = tan^2(theta):cot^2(theta).

Multiple choice maths number system representation of irrational numbers on number line existence of irrational numbers irrational numbers

Following are the steps to represent $\sqrt5$  on number line.
Arrange them in order.
1) Draw OC on line with $l(OC)=l(OB)$,
2) Draw $AB \perp OA\ and\ l(AB) =1$
3) Take $l(OA)=2$
4) $l(OC)=\sqrt5$, C is required point on real line.

  1. $1,2,3,4$
  2. $2,4,1,3$
  3. $3,2,4,1$
  4. $3,2,1,4$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The correct order of representing $\sqrt { 5 } $ on number line is 

Step 1. Take $l(OA) = 2$.
Step 2. Draw $AB$$\perp $$OA\ and\  l(AB)$ $= 1$
Step 3. Draw $OC$ on line with $l(OC) = l(OB)$
Step 4. $l(OC) =$ $\sqrt { 5 } $, $C$ is the required point on real line
Therefore, option(D) is correct.