Coordinate System and Straight Line · Mathematics · NDA
MCQ (Single Correct Answer)
1
The area of the figure formed by the lines
ax + by + c = 0, ax $$-$$ by + c = 0, ax + by $$-$$ c = 0 and ax $$-$$ by $$-$$ c = 0 is
ax + by + c = 0, ax $$-$$ by + c = 0, ax + by $$-$$ c = 0 and ax $$-$$ by $$-$$ c = 0 is
NDA 2015 Paper 2
2
If a line is perpendicular to the line 5x $$-$$ y = 0 and forms a triangle of area 5 sq units with coordinate axes, then its equation is
NDA 2015 Paper 2
3
The three lines 4x + 4y = 1, 8x $$-$$ 3y = 2, y = 0 are
NDA 2015 Paper 2
4
The line 3x + 4y $$-$$ 24 = 0 intersects the X-axis at A and Y-axis at B. Then, the circumcentre of the $$\Delta$$OAB, where O is the origin, is
NDA 2015 Paper 2
5
The product of the perpendiculars from the two points ($$\pm$$ 4, 0) to the line 3x cos$$\phi$$ + 5y sin$$\phi$$ = 15 is
NDA 2015 Paper 2
6
Two straight lines passing through the point A(3, 2) cut the line 2y = x + 3 and X-axis perpendicularly at P and Q, respectively. The equation of the line PQ is
NDA 2015 Paper 2
7
An equilateral triangle has one vertex at (0, 0) and another at (3, $$\sqrt 3 $$). What are the coordinates of the third vertex?
NDA 2016 Paper 2
8
What is the equation of the right bisector of the line segment joining (1, 1) and (2, 3) ?
NDA 2016 Paper 2
9
If the point (a, a) lies between the lines $$\left| {x + y} \right| = 2$$, then which one of the following is correct?
NDA 2016 Paper 2
10
What is the equation of the straight line which passes through the point of intersection of the straight lines x + 2y = 5 and 3x + 7y = 17 and is perpendicular to the straight line 3x + 4y = 10?
NDA 2016 Paper 2
11
If (a, b) is at unit distance from the line 8x + 6y + 1 = 0, then which of the following conditions are correct?
I. 3a $$-$$ 4b $$-$$ 4 = 0
II. 8a + 6b + 11 = 0
III. 8a + 6b $$-$$ 9 = 0
Select the correct answer using the code given below
I. 3a $$-$$ 4b $$-$$ 4 = 0
II. 8a + 6b + 11 = 0
III. 8a + 6b $$-$$ 9 = 0
Select the correct answer using the code given below
NDA 2016 Paper 2
12
A straight line cuts off an intercept of 2 units on the positive direction of X-axis and passes through the point ($$-$$3, 5). What is the foot of the perpendicular drawn from the point (3, 3) on this line?
NDA 2016 Paper 2
13
What is the value of AC2 $$-$$ BD2 ?
NDA 2016 Paper 1
14
What is the point of intersection of the diagonals ?
NDA 2016 Paper 1
15
What is the area of the parallelogram?
NDA 2016 Paper 1
16
A straight line intersects x and y axes at P and Q, respectively. If (3, 5) is the middle point of PQ, then what is the area of the $$\Delta$$POQ?
NDA 2016 Paper 1
17
What is the equation of the line passing through the point of intersection of the given lines and parallel to X-axis?
NDA 2016 Paper 1
18
What is the equation of the line passing through the point of intersection of the given lines and parallel to Y-axis?
NDA 2016 Paper 1
19
(a, 2b) is the mid-point of the line segment joining the points (10, $$-$$ 6) and (k, 4). If a $$-$$ 2b = 7, then what is the value of k?
NDA 2016 Paper 1
20
What is the acute angle between the lines represented by the equations y $$-$$ $$\sqrt 3 $$x $$-$$ 5 = 0 and $$\sqrt 3 $$y $$-$$ x + 6 =0 ?
NDA 2016 Paper 1
21
The points (a, b), (0, 0), ($$-$$a, $$-$$b) and (ab, b2) are
NDA 2017 Paper 2
22
The angle between the lines x + y $$-$$ 3 = 0 and x $$-$$ y + 3 = 0 is $$\alpha$$ and the acute angle between the lines x $$-$$ $$\sqrt 3 $$y + 2$$\sqrt 3 $$ = 0 and $$\sqrt 3 $$x $$-$$ y + 1 = 0 is $$\beta$$. Which one of the following is correct?
NDA 2017 Paper 2
23
The distance of the point (1, 3) from the line 2x + 3y = 6, measured parallel to the line 4x + y = 4 is
NDA 2017 Paper 2
24
The equation of straight line which cuts off an intercept of 5 units on negative direction of Y-axis and makes and angle 120$$^\circ$$ with positive direction of X-axis is
NDA 2017 Paper 2
25
The equation of the line passing through the point (2, 3) and the point of intersection of lines 2x $$-$$ 3y + 7 = 0 and 7x + 4y + 2 = 0 is
NDA 2017 Paper 2
26
If a vertex of a triangle is (1, 1) and the midpoints of two sides of the triangle through this vertex are ($$-$$1, 2) and (3, 2), then the centroid of the triangle is
NDA 2017 Paper 1
27
The incentre of the triangle with vertices A(1, $$\sqrt 3 $$), B(0, 0) and C(2, 0) is
NDA 2017 Paper 1
28
If the three consecutive vertices of a parallelogram are ($$-$$2, $$-$$1), (1, 0) and (4, 3), then what are the coordinates of the fourth vertex?
NDA 2017 Paper 1
29
What is the ratio in which the point $$C\left( { - {2 \over 7}, - {{20} \over 7}} \right)$$ divides the line joining the points A($$-$$2, $$-$$2) and B(2, $$-$$4)?
NDA 2017 Paper 1
30
What is the equation of the straight line parallel to 2x + 3y + 1 = 0 and passes through the point ($$-$$1, 2) ?
NDA 2017 Paper 1
31
What is the acute angle between the pair of straight lines $$\sqrt 2 x + \sqrt 3 y = 1$$ and $$\sqrt 3 x + \sqrt 2 y = 2$$ ?
NDA 2017 Paper 1
32
If the centroid of a triangle formed by (7, x), (y, $$-$$6) and (9, 10) is (6, 3), then the values of x and y are respectively
NDA 2017 Paper 1
33
The angle between the two lines $$lx + my + n = 0$$ and $$l'x + m'y + n' = 0$$ is given by $${\tan ^{ - 1}}\theta $$. What is $$\theta$$ equal to?
NDA 2018 Paper 2
34
Consider the following statements
1. The distance between the lines $$y = mx + {c_1}$$ and $$y = mx + {c_2}$$ is $${{\left| {{c_1} - {c_2}} \right|} \over {\sqrt {1 + {m^2}} }}$$.
2. The distance between the lines $$ax + by + {c_1} = 0$$ and $$ax + by + {c_2} = 0$$ is $${{\left| {{c_1} - {c_2}} \right|} \over {\sqrt {{a^2} + {b^2}} }}$$.
3.The distance between the lines $$x = {c_1}$$ and $$x = {c_2}$$ is $$\left| {{c_1} - {c_2}} \right|$$.
Which of the above statements are correct?
1. The distance between the lines $$y = mx + {c_1}$$ and $$y = mx + {c_2}$$ is $${{\left| {{c_1} - {c_2}} \right|} \over {\sqrt {1 + {m^2}} }}$$.
2. The distance between the lines $$ax + by + {c_1} = 0$$ and $$ax + by + {c_2} = 0$$ is $${{\left| {{c_1} - {c_2}} \right|} \over {\sqrt {{a^2} + {b^2}} }}$$.
3.The distance between the lines $$x = {c_1}$$ and $$x = {c_2}$$ is $$\left| {{c_1} - {c_2}} \right|$$.
Which of the above statements are correct?
NDA 2018 Paper 2
35
What is the equation of straight line passing through the point of intersection of the lines $${x \over 2} + {y \over 3} = 1$$ and $${x \over 3} + {y \over 2} = 1$$ and parallel to the line $$4x + 5y - 6 = 0$$ ?
NDA 2018 Paper 2
36
What is the distance between the points which divide the line segment joining (4, 3) and (5, 7) internally and externally in the ratio 2 : 3 ?
NDA 2018 Paper 1
37
What is the angle between the straight lines $$({m^2} - mn)y = (mn + {n^2})x + {n^3}$$ and $$(mn + {m^2})y = (mn - {n^2})x + {m^3}$$, where m > n?
NDA 2018 Paper 1
38
What is the equation of the straight line cutting-off an intercept 2 from the negative direction of Y-axis and inclined at 30$$^\circ$$ with the positive direction of X-axis?
NDA 2018 Paper 1
39
What is the equation of the line passing through the point of intersection of the lines $$x + 2y - 3 = 0$$ and $$2x - y + 5 = 0$$ and parallel to the line $$y - x + 10 = 0$$ ?
NDA 2018 Paper 1
40
Consider the following statements
I. The length p of the perpendicular from the origin to the line ax + by = c satisfies the relation $${p^2} = {{{c^2}} \over {{a^2} + {b^2}}}$$.
II. The length p of the perpendicular from the origin to the line $${x \over a} + {y \over b} = 1$$ satisfied the relation $${1 \over {{p^2}}} = {1 \over {{a^2}}} + {1 \over {{b^2}}}$$.
III. The length p of the perpendicular from the origin to the line y = mx + c satisfies the relation $${1 \over {{p^2}}} = {{1 + {m^2} + {c^2}} \over {{c^2}}}$$.
Which of the above is/are correct?
I. The length p of the perpendicular from the origin to the line ax + by = c satisfies the relation $${p^2} = {{{c^2}} \over {{a^2} + {b^2}}}$$.
II. The length p of the perpendicular from the origin to the line $${x \over a} + {y \over b} = 1$$ satisfied the relation $${1 \over {{p^2}}} = {1 \over {{a^2}}} + {1 \over {{b^2}}}$$.
III. The length p of the perpendicular from the origin to the line y = mx + c satisfies the relation $${1 \over {{p^2}}} = {{1 + {m^2} + {c^2}} \over {{c^2}}}$$.
Which of the above is/are correct?
NDA 2018 Paper 1
41
What is the equation of the straight line passing through the point (2, 3) and making an intercept on the positive Y-axis equal to twice its intercept on the positive X-axis?
NDA 2018 Paper 1
42
The perpendiculars that fall from any point of the straight line 2x + 11y = 5 upon the two straight lines 24x + 7y = 20 and 4x $$-$$ 3y = 2 are
NDA 2018 Paper 1
43
The equation of the line, when the portion of it intercepted between the axes is divided by the point (2, 3) in the ratio of 3 : 2, is
NDA 2018 Paper 1
44
What is the distance between the straight lines 3x + 4y = 9 and 6x + 8y = 15 ?
NDA 2018 Paper 1
45
What is the distance between the points $$P(m\cos 2\alpha ,m\sin 2\alpha )$$ and $$Q(m\cos 2\beta ,m\sin 2\beta )$$ ?
NDA 2019 Paper 2
46
An equilateral triangle has one vertex at ($$-$$1, $$-$$1) and another vertex at $$( - \sqrt 3 ,\sqrt 3 )$$. The third vertex may lie on
NDA 2019 Paper 2
47
The equation ax + by + c = 0 represents a straight line
NDA 2019 Paper 2
48
What is the angle between the lines x cos$$\alpha$$ + y sin$$\alpha$$ = a and x sin$$\beta$$ $$-$$ y cos$$\beta$$ = a ?
NDA 2019 Paper 2
49
Consider the following statements
1. For an equation of a line, $$x\cos \theta + y\sin \theta = p$$, in normal form, the length of the perpendicular from the point ($$\alpha$$, $$\beta$$) to the line is $$\left| {\alpha \cos \theta + \beta \sin \theta + p} \right|$$.
The length of the perpendicular from the point ($$\alpha$$, $$\beta$$) to the line $${x \over a} + {y \over b} = 1$$ is $$\left| {{{a\alpha + b\beta - ab} \over {\sqrt {{a^2} + {b^2}} }}} \right|$$.
Which of the above statements is/are correct?
1. For an equation of a line, $$x\cos \theta + y\sin \theta = p$$, in normal form, the length of the perpendicular from the point ($$\alpha$$, $$\beta$$) to the line is $$\left| {\alpha \cos \theta + \beta \sin \theta + p} \right|$$.
The length of the perpendicular from the point ($$\alpha$$, $$\beta$$) to the line $${x \over a} + {y \over b} = 1$$ is $$\left| {{{a\alpha + b\beta - ab} \over {\sqrt {{a^2} + {b^2}} }}} \right|$$.
Which of the above statements is/are correct?
NDA 2019 Paper 1
50
The points (1, 3) and (5, 1) are two opposite vertices of a rectangle. The other two vertices lie on the line y = 2x + c. What is the value of c?
NDA 2019 Paper 1
51
If the lines 3y + 4x = 1, y = x + 5 and 5y + bx = 3 are concurrent, then what is the value of b?
NDA 2019 Paper 1
52
What is the equation of the straight line which is perpendicular to y = x and passes through (3, 2) ?
NDA 2019 Paper 1
53
The straight lines x + y $$-$$ 4 = 0, 3x + y $$-$$ 4 = 0 and x + 3y $$-$$ 4 = 0 form a triangle, which is
NDA 2019 Paper 1
54
The number of points represented by the equation $x = 5$ on the $xy$-plane is
NDA Mathematics 21 April 2024
55
If a variable line passes through the point of intersection of the lines $x + 2y - 1 = 0$ and $2x - y - 1 = 0$ and meets the coordinate axes in $A$ and $B$, then what is the locus of the mid-point of $AB$?
NDA Mathematics 21 April 2024
56
What is the equation to the straight line passing through the point $(-sin\theta, cos\theta)$ and perpendicular to the line $xcos\theta + ysin\theta = 9$?
NDA Mathematics 21 April 2024
57
Two points $P$ and $Q$ lie on line $y = 2x + 3$. These two points $P$ and $Q$ are at a distance 2 units from another point $R(1, 5)$. What are the coordinates of the points $P$ and $Q$?
NDA Mathematics 21 April 2024
58
If two sides of a square lie on the lines $2x + y - 3 = 0$ and $4x + 2y + 5 = 0$, then what is the area of the square in square units?
NDA Mathematics 21 April 2024
59
ABC is a triangle with A(3, 5). The mid-points of sides AB, AC are at (-1, 2), (6, 4) respectively. What are the coordinates of centroid of the triangle ABC?
NDA Mathematics 21 April 2024
60
ABC is an acute angled isosceles triangle. Two equal sides AB and AC lie on the lines 7x - y - 3 = 0 and x + y - 5 = 0. If θ is one of the equal angles, then what is cotθ equal to?
NDA Mathematics 21 April 2024
61
Let $x+2 y+1=0$ and $2 x+3 y+4=0$ are two lines of regression computed from some bivariate data. If $\theta$ is the acute angle between them, then what is the value of $488 \tan 3 \theta$ ?
NDA Mathematics 21 April 2024
62
The points (-a, -b), (0, 0), (a, b) and (a2, ab) are:
NDA Mathematics 3 September 2023
63
Given that 16p2 + 49q2 - 4r2 - 56pq = 0. Which one of the following is a point on a pair of straight lines (px + qy + r) (px + qy - r) = 0?
NDA Mathematics 3 September 2023
64
For what values of k is the line (k - 3)x - (5 - k2) y + k2 - 7k + 6 = 0 parallel to the line x + y = 1?
NDA Mathematics 3 September 2023
65
The line x + y = 4 cuts the line joining P(-1, 1) and Q(5, 7) at R. What is PR ∶ RQ equal to ?
NDA Mathematics 3 September 2023
66
What is the sum of the intercepts of the line whose perpendicular distance from origin is 4 units and the angle which the normal makes with positive direction of x-axis is 15°?
NDA Mathematics 3 September 2023
67
What is the equation of diagonal through origin ?
NDA Mathematics 16 April 2023
68
What is the equation of other diagonal ?
NDA Mathematics 16 April 2023
69
If two lines of regression are x + 4y + 1 = 0 and 4x + 9y + 7 = 0, then what is the value of x when y = -3 ?
NDA Mathematics 16 April 2023
70
A bivariate data set contains only two points (-1, 1) and (3, 2). What will be the line of regression of y on x ?
NDA Mathematics 16 April 2023
71
Consider the following statements in respect of the line passing through origin and inclining at an angle of 75° with the positive direction of x-axis :
1. The line passes through the point $\left(1, \frac{1}{2−\sqrt{3}}\right)$.
2. The line entirely lies in first and third quadrants.
Which of the statements given above is/are correct ?
NDA Mathematics 4 September 2022
72
If P(3, 4) is the mid-point of a line segment between the axes, then what is the equation of the line?
NDA Mathematics 4 September 2022
73
The base AB of an equilateral triangle ABC with side 8 cm lies along the y-axis such that the mid-point of AB is at the origin and B lies above the origin. What is the equation of line passing through (8, 0) and parallel to the side AC ?
NDA Mathematics 4 September 2022
74
Consider the inequations 5x - 4y + 12 < 0, x + y < 2, x < 0 and y > 0. Which one of the following points lies in the common region?
NDA Mathematics 10 April 2022
75
If the points with coordinates (-5, 0), (5p2, 10p) and (5q2, 10q) are collinear, then what is the value of pq where p ≠ q ?
NDA Mathematics 10 April 2022
76
What is the equation of the straight line which passes through the point (1, -2) and cuts off equal intercepts from the axes ?
NDA Mathematics 10 April 2022
77
A straight line passes through the point of intersection of x + 2y + 2 = 0 and 2x - 3y - 3 = 0. It cuts equal intercepts in the fourth quadrant. What is the sum of the absolute values of the intercepts?
NDA Mathematics 10 April 2022
78
Under which one of the following conditions are the lines ax + by + c = 0 and bx + ay + c = 0 parallel (a ≠ 0, b ≠ 0)?
NDA Mathematics 10 April 2022
79
What is the equation of the locus of the mid-point of the line segment obtained by cutting the line x + y = p, (where p is a real number) by the coordinate axes ?
NDA Mathematics 10 April 2022
80
If the point (x, y) is equidistant from the points (2a, 0) and (0, 3a) where a > 0, then which one of the following is correct ?
NDA Mathematics 10 April 2022
81
A parallelogram has three consecutive vertices (-3, 4), (0, -4) and (5, 2). The fourth vertex is
NDA Mathematics 18 April 2021
82
If the lines y + px = 1 and y - qx = 2 are perpendicular, then which one of the following is correct?
NDA Mathematics 18 April 2021
83
If A, B and C are in AP, then the straight line Ax + 2By + C = 0 will always pass through a fixed point. The fixed point is
NDA Mathematics 18 April 2021
84
If the image of the point (-4, 2) by a line mirror is (4, -2), then what is the equation of the line mirror?
NDA Mathematics 18 April 2021
85
Consider the following statements in respect of the points (p, p - 3), (q + 3, q) and (6, 3):
1. The points lie on a straight line.
2. The points always lie in the first quadrant only for any value of p and q.
Which of the above statements is/are correct?
NDA Mathematics 18 April 2021
86
What is the acute angle between the lines x - 2 = 0 and √3x - y - 2 = 0?
NDA Mathematics 18 April 2021
87
The point of intersection of diagonals of a square ABCD is at the origin and one of its vertices is at A(4, 2). What is the equation of the diagonal BD?
NDA Mathematics 18 April 2021