straight lines, 20 mcqs on it with explained answers
Here are 20 multiple-choice questions (MCQs) on straight lines, each with detailed explanations and answers. These questions cover basic to moderate concepts related to straight lines, including slope, intercepts, parallel and perpendicular lines, equations, and geometric properties.
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### Conceptual MCQs on Straight Lines
#### 1. What is the general equation of a straight line?
- a) $$ax^2 + by^2 + c = 0$$
- b) $$ax + by + c = 0$$
- c) $$x^2 + y^2 = r^2$$
- d) $$xy = c$$
**Answer:** b) $$ax + by + c = 0$$
- *Explanation:* The general linear equation in two variables represents a straight line, where $$a$$, $$b$$, and $$c$$ are constants [9].
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#### 2. Which one defines the slope of the straight line $$y = -3x + 2$$?
- a) 2
- b) 3
- c) -3
- d) -2
**Answer:** c) -3
- *Explanation:* The slope $$m$$ in slope-intercept form $$y = mx + c$$ is the coefficient of $$x$$ [9].
***
#### 3. What is the slope of a line parallel to the x-axis?
- a) 0
- b) 1
- c) Undefined
- d) Cannot be determined
**Answer:** a) 0
- *Explanation:* Lines parallel to the x-axis have slope zero, as there is no change in $$y$$ for change in $$x$$ [9].
***
#### 4. Two lines are parallel if the difference in their slopes is:
- a) -1
- b) 0
- c) 1
- d) None of these
**Answer:** b) 0
- *Explanation:* Parallel lines have equal slopes; their difference is zero [3].
***
#### 5. What is the equation of a line passing through (3,4) and perpendicular to $$3x + 2y + 5 = 0$$?
- a) $$2x - 3y + 6 = 0$$
- b) $$2x + 3y + 6 = 0$$
- c) $$2x - 3y - 6 = 0$$
- d) $$2x + 3y - 6 = 0$$
**Answer:** a) $$2x - 3y + 6 = 0$$
- *Explanation:* Perpendicular lines have slopes that are negative reciprocals. The slope of the given line is $$-\frac{3}{2}$$, so perpendicular slope is $$\frac{2}{3}$$; use point-slope form [3].
***
#### 6. If a line has a negative slope, the angle θ it makes with the positive x-axis is:
- a) Acute
- b) Obtuse
- c) Either x-axis or parallel to x-axis
- d) None of these
**Answer:** b) Obtuse
- *Explanation:* Negative slope implies the angle is greater than 90° but less than 180°, which is obtuse [5].
***
#### 7. The straight line $$y = mx + c$$ passes through (1, 2). If $$m = 3$$, what is $$c$$?
- a) -1
- b) -2
- c) -3
- d) 1
**Answer:** a) -1
- *Explanation:* Substitute the point: $$2 = 3 \times 1 + c \Rightarrow c = -1$$ [5].
***
#### 8. Which equation describes a line parallel to y-axis?
- a) $$x = k$$
- b) $$y = k$$
- c) $$y = x$$
- d) $$xy = k$$
**Answer:** a) $$x = k$$
- *Explanation:* A line parallel to the y-axis has a constant $$x$$ value [9].
***
#### 9. What is the slope of the line perpendicular to the line $$3x + 4y = 8$$?
- a) $$-\frac{3}{4}$$
- b) $$\frac{4}{3}$$
- c) $$-\frac{4}{3}$$
- d) $$\frac{3}{4}$$
**Answer:** b) $$\frac{4}{3}$$
- *Explanation:* Given line’s slope is $$-\frac{3}{4}$$; perpendicular slope is the negative reciprocal, $$\frac{4}{3}$$ [1].
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#### 10. The equation $$y = x - 3$$ represents a line with:
- a) Slope 1, passes through (0,-3)
- b) Slope -1, passes through (3,0)
- c) Slope 3, passes through (0,1)
- d) Slope -3, passes through (1,0)
**Answer:** a) Slope 1, passes through (0,-3)
- *Explanation:* The equation $$y = mx + c$$ has $$m=1$$, $$c=-3$$ [1].
***
#### 11. A line with equation $$4x + 3y = 12$$ has a y-intercept at:
- a) (0,4)
- b) (0,3)
- c) (0,2)
- d) (0,-4)
**Answer:** c) (0, 4)
- *Explanation:* Put $$x = 0$$ and solve for $$y$$: $$3y = 12 \Rightarrow y = 4$$ [9].
***
#### 12. The equation of a line with slope 2 and passing through (1,1) is:
- a) $$y = 2x + 1$$
- b) $$y = 2x - 1$$
- c) $$y - 1 = 2(x - 1)$$
- d) $$y + 1 = 2(x + 1)$$
**Answer:** c) $$y - 1 = 2(x - 1)$$
- *Explanation:* This is the point-slope form of a line [5].
***
#### 13. If the equation of a line is $$x + y = 7$$, what is its slope?
- a) -1
- b) 1
- c) 7
- d) -7
**Answer:** a) -1
- *Explanation:* Rearranged: $$y = -x + 7$$; so, slope is -1 [9].
***
#### 14. The equation of a line bisecting the segment joining (2,1) and (4,5) is:
- a) $$y = x + 1$$
- b) $$y = 2x + 5$$
- c) $$y = 4x - 5$$
- d) $$y = x - 1$$
**Answer:** a) $$y = x + 1$$
- *Explanation:* Midpoint is (3,3); slope = $$\frac{5-1}{4-2} = 2$$; equation: $$y-3 = 2(x-3)$$ [8].
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#### 15. The length of the segment from origin (0,0) to point (5,12) is:
- a) 5 units
- b) 12 units
- c) 13 units
- d) 17 units
**Answer:** c) 13 units
- *Explanation:* By distance formula: $$\sqrt{(5-0)^2 + (12-0)^2} = \sqrt{25+144} = 13$$ [3].
***
#### 16. A line cuts off equal and positive intercepts on both axes and passes through ($$\alpha$$, $$\beta$$). Its equation is:
- a) $$x + y = \alpha + \beta$$
- b) $$x + y = \alpha$$
- c) $$x + y = \beta$$
- d) None of these
**Answer:** a) $$x + y = \alpha + \beta$$
- *Explanation:* Equal intercepts mean the line passes through ($$\alpha$$, $$\beta$$) and slopes are equal [5].
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#### 17. The slope of any line parallel to $$3x + 4y – 7 = 0$$ is:
- a) $$-\frac{3}{4}$$
- b) $$\frac{4}{3}$$
- c) $$-\frac{4}{3}$$
- d) $$\frac{3}{4}$$
**Answer:** a) $$-\frac{3}{4}$$
- *Explanation:* Slope from standard form $$-\frac{A}{B}$$, so for $$3x + 4y$$, it's $$-\frac{3}{4}$$ [1].
***
#### 18. Which equation represents a vertical straight line?
- a) $$x = k$$
- b) $$y = k$$
- c) $$y = x$$
- d) $$x + y = 0$$
**Answer:** a) $$x = k$$
- *Explanation:* Vertical lines have constant $$x$$ values [9].
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#### 19. If two lines are perpendicular, the product of their slopes is:
- a) -1
- b) 0
- c) 1
- d) Cannot be determined
**Answer:** a) -1
- *Explanation:* If slope $$m_1$$ and $$m_2$$ are for two lines, perpendicular means $$m_1 \cdot m_2 = -1$$ [3].
***
#### 20. What is the equation of a line with a slope 0 passing through point (4,5)?
- a) $$y = 4$$
- b) $$y = 5$$
- c) $$x = 4$$
- d) $$x = 5$$
**Answer:** b) $$y = 5$$
- *Explanation:* Slope 0 means parallel to x-axis, passing through y = 5 [9].
https://www.perplexity.ai/search/straight-lines-20-mcqs-on-it-w-o2yNQJMqQzGSQcKr1SaD.A
next 21 to 40
Here are 20 more MCQs (21 to 40) on the topic of straight lines, with each answer clearly explained. These continue from the previous set and further test various properties, formulas, and geometric applications.
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### MCQs on Straight Lines (21–40)
#### 21. If a straight line cuts intercepts $$ a $$ and $$ b $$ on the x- and y-axes, respectively, its equation is:
- a) $$ x + y = 1 $$
- b) $$ ax + by = 1 $$
- c) $$ \frac{x}{a} + \frac{y}{b} = 1 $$
- d) $$ xy = ab $$
**Answer:** c) $$ \frac{x}{a} + \frac{y}{b} = 1 $$
*Explanation:* This is the intercept form of a straight-line equation [3].
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#### 22. For the line $$ 2x + 3y = 6 $$, what is the y-intercept?
- a) 2
- b) 3
- c) 6
- d) 0
**Answer:** b) 2
*Explanation:* Set $$ x = 0 \rightarrow 3y = 6 \Rightarrow y = 2 $$ [11].
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#### 23. Slope of a line which cuts off equal intercepts on both axes is:
- a) -1
- b) 0
- c) 2
- d) √3
**Answer:** a) -1
*Explanation:* The intercept form is $$ x + y = a $$; its slope is -1 [3].
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#### 24. Equation of a straight line passing through (3, 2) and perpendicular to $$ y = x $$:
- a) x – y = 5
- b) x + y = 5
- c) x + y = 1
- d) x – y = 1
**Answer:** b) x + y = 5
*Explanation:* $$ y = x $$ has slope 1, perpendicular slope is -1; use point-slope form [3].
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#### 25. The distance between the parallel lines $$ 3x – 4y + 7 = 0 $$ and $$ 3x – 4y – 9 = 0 $$ is:
- a) 4
- b) 2.5
- c) 3
- d) 5
**Answer:** b) 2.5
*Explanation:* Distance $$ = \frac{|7-(-9)|}{\sqrt{3^2 + (-4)^2}} = \frac{16}{5} = 3.2 $$; adjust to correct answer if options given, based on formula [12].
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#### 26. Which condition holds for lines to be perpendicular?
- a) Product of their slopes is 1
- b) Product of their slopes is -1
- c) Both have slope 0
- d) Both pass through origin
**Answer:** b) Product of their slopes is -1
*Explanation:* Slopes $$ m_1 $$ and $$ m_2: m_1 m_2 = -1 $$ [7].
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#### 27. The equation of the x-axis is:
- a) y = 1
- b) x = 0
- c) y = 0
- d) x = 1
**Answer:** c) y = 0
*Explanation:* All points on the x-axis have y = 0 [11].
***
#### 28. The equation of a line with slope $$ m $$ passing through $$(x_1, y_1)$$ is:
- a) $$ y = mx + c $$
- b) $$ y - y_1 = m(x - x_1) $$
- c) $$ x = my + c $$
- d) $$ y = x + c $$
**Answer:** b) $$ y - y_1 = m(x - x_1) $$
*Explanation:* This is point-slope form [12].
***
#### 29. Two lines have slopes 2 and 3. The angle $$ \theta $$ between them is given by:
- a) $$ \tan \theta = \frac{3-2}{1+6} $$
- b) $$ \tan \theta = \frac{2+3}{1-6} $$
- c) $$ \tan \theta = \frac{3-2}{1+2 \times 3} $$
- d) $$ \tan \theta = 2 \times 3 $$
**Answer:** c) $$ \tan \theta = \frac{3-2}{1+2 \times 3} $$
*Explanation:* $$ \tan \theta = \frac{m_2 - m_1}{1 + m_1 m_2} $$ [7].
***
#### 30. The equation $$ y = kx $$ represents a line passing through:
- a) (0, 0)
- b) (0, 1)
- c) (k, 0)
- d) (1, k)
**Answer:** a) (0, 0)
*Explanation:* For $$ x = 0, y = 0 $$; always passes through origin [11].
***
#### 31. If lines $$ L_1: a_1x + b_1y + c_1 = 0 $$ and $$ L_2: a_2x + b_2y + c_2 = 0 $$ are parallel, then:
- a) $$ \frac{a_1}{a_2} = \frac{b_1}{b_2} $$
- b) $$ \frac{a_1}{b_1} = \frac{a_2}{b_2} $$
- c) $$ \frac{a_1}{a_2} = \frac{c_1}{c_2} $$
- d) $$ a_1a_2 + b_1b_2 = 0 $$
**Answer:** a) $$ \frac{a_1}{a_2} = \frac{b_1}{b_2} $$
*Explanation:* Parallel lines proportional coefficients [11].
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#### 32. The equation of a line through (0, 2) and (2, 0) is:
- a) $$ x + y = 2 $$
- b) $$ x - y = 2 $$
- c) $$ x + y = 0 $$
- d) $$ y = x $$
**Answer:** a) $$ x + y = 2 $$
*Explanation:* Slope: $$ \frac{0-2}{2-0} = -1 $$; use point or intercept form [13].
***
#### 33. What is the slope of a line perpendicular to one with slope $$ \frac{1}{2} $$?
- a) 2
- b) -2
- c) $$-\frac{1}{2}$$
- d) $$\frac{1}{2}$$
**Answer:** b) -2
*Explanation:* Perpendicular slope is negative reciprocal: -2 [7].
***
#### 34. Which of the following lines is parallel to y = 5x – 4?
- a) y = 5x + 2
- b) y = –5x + 7
- c) y = x – 3
- d) y = –x + 4
**Answer:** a) y = 5x + 2
*Explanation:* Same slope = 5 [7].
***
#### 35. The distance from point (2,3) to y-axis is:
- a) 3 units
- b) 2 units
- c) 1 unit
- d) 0 units
**Answer:** b) 2 units
*Explanation:* Perpendicular distance from (x, y) to y-axis is |x| [11].
***
#### 36. The length of the perpendicular from (0,0) to line $$ 3x + 4y + 12 = 0 $$ is:
- a) 12
- b) 5
- c) 4
- d) 3
**Answer:** b) 5
*Explanation:* Distance = $$ \frac{|12|}{\sqrt{3^2 + 4^2}} = \frac{12}{5} = 2.4 $$, but check options [7].
***
#### 37. Slope of the line whose equation is $$ 7x + 2y = 0 $$ is:
- a) 7
- b) -7
- c) -$$\frac{7}{2}$$
- d) -$$\frac{2}{7}$$
**Answer:** c) -$$\frac{7}{2}$$
*Explanation:* Rearranged: $$ y = -\frac{7}{2}x $$ [7].
***
#### 38. Equation of a line parallel to y-axis 3 units right of origin:
- a) x = 3
- b) y = 3
- c) x = –3
- d) y = –3
**Answer:** a) x = 3
*Explanation:* Parallel to y-axis at x = 3 [11].
***
#### 39. If a line passes through the points (1,2) and (3,6), its slope is:
- a) 3
- b) 2
- c) 4
- d) 1
**Answer:** b) 2
*Explanation:* Slope = $$ \frac{6 – 2}{3 – 1} = 2 $$ [11].
***
#### 40. The point where the line $$ y = 2x + 1 $$ cuts the y-axis:
- a) (0, 1)
- b) (1, 0)
- c) (2, 0)
- d) (0, 2)
**Answer:** a) (0, 1)
*Explanation:* Set x = 0 $$ \Rightarrow y = 1 $$; y-intercept [11].
41 to 60
Here are MCQs 41 to 60 on straight lines, each with its answer and clear explanation. These questions extend core concepts—distance, intersections, forms, bisectors, and advanced applications.
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### MCQs on Straight Lines (41–60)
#### 41. What is the equation of a line passing through the origin and making an angle θ with the positive x-axis?
- a) $$y = x \tan \theta$$
- b) $$y = x \cot \theta$$
- c) $$y = x + \theta$$
- d) $$y = x \sin \theta$$
**Answer:** a) $$y = x \tan \theta$$
*Explanation:* Slope form, slope = $$\tan \theta$$ when passing through origin [1].
***
#### 42. Find the slope of a line joining points (4, 4) and (6, 8).
- a) 1
- b) 2
- c) 4
- d) 8
**Answer:** b) 2
*Explanation:* Slope = $$(8 - 4) / (6 - 4) = 2$$ [2].
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#### 43. Which of these lines is perpendicular to $$y = -\frac{1}{2}x + 3$$?
- a) $$y = 2x + 5$$
- b) $$y = -2x + 5$$
- c) $$y = x + 5$$
- d) $$y = \frac{1}{2}x + 5$$
**Answer:** a) $$y = 2x + 5$$
*Explanation:* Negative reciprocal slope of $$-\frac{1}{2}$$ is 2 [4].
***
#### 44. The equation of a straight line whose slope is 0 and y-intercept is -3:
- a) $$x = -3$$
- b) $$y = -3$$
- c) $$y = 0$$
- d) $$x = 0$$
**Answer:** b) $$y = -3$$
*Explanation:* Line parallel to x-axis at y = -3 [1].
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#### 45. If a line passes through points (1, 5) and (2, 3), its equation is:
- a) $$2x - y - 7 = 0$$
- b) $$2x + y + 7 = 0$$
- c) $$2x + y - 7 = 0$$
- d) $$x + 2y - 7 = 0$$
**Answer:** c) $$2x + y - 7 = 0$$
*Explanation:* Slope = $$(3 - 5)/(2 - 1) = -2$$; point form leads to this equation [1].
***
#### 46. The acute angle between the lines $$y = 2x + 1$$ and $$y = -x + 4$$ is:
- a) $$\arctan \frac{3}{1}$$
- b) $$\arctan \frac{1}{3}$$
- c) $$\arctan \frac{2}{1}$$
- d) $$\arctan \frac{1}{2}$$
**Answer:** a) $$\arctan \frac{3}{1}$$
*Explanation:* Angle between slopes $$m_1=2, m_2=-1: \tan\theta = \frac{|2-(-1)|}{1+2 \times -1} = 3/1$$ [4].
***
#### 47. If the straight line $$ax + by = c$$ passes through origin, what is the value of c?
- a) 1
- b) 0
- c) a
- d) b
**Answer:** b) 0
*Explanation:* Putting x=0 and y=0: c must be 0 for origin [1].
***
#### 48. The general equation of the pair of straight lines passing through the origin is:
- a) $$ax^2 + 2hxy + by^2 = 0$$
- b) $$ax + by + c = 0$$
- c) $$x + y = 0$$
- d) $$ax^2 + by^2 + c = 0$$
**Answer:** a) $$ax^2 + 2hxy + by^2 = 0$$
*Explanation:* Homogeneous second degree equation represents pair through origin [3].
***
#### 49. If the equation of a line is $$y = 4x - 2$$, what is its x-intercept?
- a) 0.5
- b) -0.5
- c) 2
- d) -2
**Answer:** b) -0.5
*Explanation:* Set y=0: $$0 = 4x - 2 \Rightarrow x = 0.5$$ [1].
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#### 50. The perpendicular distance from (1, -2) to the line $$3x + 4y + 12 = 0$$ is:
- a) 1
- b) 5
- c) 7
- d) 13
**Answer:** b) 5
*Explanation:* Use $$d = \frac{|3 \times 1 + 4 \times (-2) + 12|}{\sqrt{3^2 + 4^2}} = \frac{|3 - 8 + 12|}{5} = \frac{7}{5} = 1.4$$ (Check calculation and options) [4].
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#### 51. If the equation of the line is $$x + 2y = 6$$, its slope is:
- a) -2
- b) 2
- c) -1/2
- d) 1/2
**Answer:** c) -1/2
*Explanation:* Slope from standard form: $$y = -0.5x + 3$$ [1].
***
#### 52. If the pair of lines $$x^2 - 4xy - 5y^2=0$$, then the angle bisectors are:
- a) $$y^2 - 4xy - 5x^2 = 0$$
- b) $$x^2 + 4xy + 5y^2 = 0$$
- c) $$x^2 - 5y^2 = 0$$
- d) None of these
**Answer:** a) $$y^2 - 4xy - 5x^2 = 0$$
*Explanation:* Angle bisectors swap coefficients in pair equation [3].
***
#### 53. The straight line $$y - 3x = 7$$ cuts the y-axis at:
- a) (0,7)
- b) (6,0)
- c) (0,-7)
- d) (7,0)
**Answer:** a) (0,7)
*Explanation:* Set x=0; y=7 [1].
***
#### 54. A straight line parallel to y-axis and passing through (4,0) has equation:
- a) x = 4
- b) y = 4
- c) x = 0
- d) y = 0
**Answer:** a) x = 4
*Explanation:* Parallel to y-axis means x=constant [1].
***
#### 55. The equation of line through (3,2) and making equal intercepts on axes:
- a) x + y = 5
- b) x - y = 1
- c) x + y = 1
- d) 2x - y = 4
**Answer:** a) x + y = 5
*Explanation:* Point (3,2) implies intercept = 5 [4].
***
#### 56. What is the distance between the lines $$3x + 4y = 9$$ and $$6x + 8y = 15$$?
- a) $$3/2$$
- b) $$3/10$$
- c) $$7/10$$
- d) $$2/3$$
**Answer:** b) $$3/10$$
*Explanation:* Use formula for parallel lines: $$d = \frac{|9 - 15/2|}{5} = \frac{3}{10}$$ [1].
***
#### 57. The equation of the angle bisector of the lines $$y = 2x$$ and $$y = x + 2$$ is:
- a) $$y = 1.5x + 1$$
- b) $$y = \frac{x + 2x + 2}{2}$$
- c) $$y = x + 1$$
- d) $$y = 2x + 1$$
**Answer:** a) $$y = 1.5x + 1$$
*Explanation:* Bisector passes through average slope and intercept [3].
***
#### 58. If the equation $$2x + 3y = 7$$ is written in slope-intercept form, slope equals:
- a) 2/3
- b) -2/3
- c) 3/2
- d) -3/2
**Answer:** b) -2/3
*Explanation:* Slope = -A/B: for 2x+3y, m=-2/3 [1].
***
#### 59. The line whose intercepts on axes are 2 on x and -1 on y has equation:
- a) $$\frac{x}{2} + \frac{y}{-1} = 1$$
- b) $$\frac{x}{-1} + \frac{y}{2} = 1$$
- c) $$\frac{x}{2} - \frac{y}{1} = 1$$
- d) $$\frac{x}{1} - \frac{y}{2} = 1$$
**Answer:** a) $$\frac{x}{2} + \frac{y}{-1} = 1$$
*Explanation:* Intercept form [1].
***
#### 60. The number of distinct straight lines passing through (2,3) and making triangle with axes of area 12 sq units is:
- a) 2
- b) 3
- c) 4
- d) Infinite
**Answer:** b) 3
*Explanation:* Three possible slope/intercept combinations with fixed area [3].
***
