Mathematics · Quantitative Aptitude

Coordinate Geometry and Construction

104 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 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

A straight line through the origin O meets the parallel lines 4x+2y=9 and 2x+y+6=0 at point P and Q respectively. Then the point O divides the segment PQ in the ratio

  1. 1:2

  2. 3:4

  3. 2:1

  4. 4:3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The parallel lines are 4x + 2y - 9 = 0 (or normalized as 2x + y - 4.5 = 0) and 2x + y + 6 = 0. A line through the origin has the form y = mx. It intersects the first line at P and the second line at Q. The distances from the origin to the lines along any ray are proportional to the constant terms of the parallel lines. Specifically, the ratio OP/OQ equals the ratio of the constant distances from the origin, which is 4.5 / 6 = 9/12 = 3/4.

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 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

A point which divides the joint of $(1,2)$ and $(3,4)$ externally in the ratio $1:1$

  1. Lies in the first quadrant

  2. Lies in the second quadrant

  3. Lies in third quadrat

  4. Cannot be found
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The section formula for external division in the ratio m:n uses the formula ((mx2 - nx1)/(m - n), (my2 - ny1)/(m - n)). When the ratio is 1:1, the denominator becomes m - n = 1 - 1 = 0, which results in division by zero. Therefore, a point dividing a line segment externally in the ratio 1:1 cannot be found mathematically.

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

If the ratio in which the line segment joining the points (6,4) and (x,-7) divided internally by y-axis is 6: 1, then x equals

  1. 2

  2. 3

  3. -1

  4. -2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The y-axis divides a line segment in the ratio -x1:x2 or by setting the x-coordinate of the section formula to zero. Using the given coordinates (6, 4) and (x, -7) with a ratio of 6:1, the x-coordinate formula gives (6*x + 1*6)/(6 + 1) = 0, which yields 6x + 6 = 0, so x = -1.

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.

Multiple choice maths construction of polygons construction of parallelograms and rectangles construction of special quadrilaterals constructions related to a quadrilateral

Construct a rectangle $ABCD$, where $AB=10$ cm and $BC=8$ cm.Steps for its construction is given in a jumbled form. Identify its correct sequence.
1) Join these cuts with a line $CD$ and rectangle $ABCD$ is formed
2) Draw a straight line $AB$ of length $10$ cm
3) Draw perpendicular lines at $A$ and $B$ using protractor.
4) Using compass cut arc at the perpendicular from $A$ and $B$ of lengths $8$ cm

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

Correct sequence for constructing rectangle $ABCD$ is:

Draw a straight line $AB$ of $10 $ cm.
Draw perpendicular lines at $A$ and $B$ using proctor.
Using compass cut arc at the perpendicular from $A$ and $B$ of lengths $8$ cm.
Join these cuts with a line $CD$ and rectangle $ABCD$ is formed.
Correct sequence is $2,3,4,1$.

Multiple choice maths how much does it weigh? basic operations with same units operations involving units of length calculations define weight and units of weight

The ratio at which the point $(5,4)$ divides the line $(3,2)$ and $(8,7)$

  1. $\dfrac 23$
  2. $\dfrac{-3}{2}$
  3. $\dfrac{1}{2}$
  4. $\dfrac{1}{9}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given points $(3,2);(8,7)$
Let the ratio be $m:n$
The dividing point is given as $\dfrac{8m+3n}{m+n}=5\\8m+3n=5m+5n\\3m=2n\\\dfrac mn=\dfrac 23$