Question:
The graph of which of the following equations is a straight line that is parallel to line M in the figure above and intersects the negative direction of Y-axis?
Indicate all such equations.
[Note: Select one or more answer choices]
\includegraphics[width=0.7\linewidth]{04.png}
The graph of which of the following equations is a straight line that is parallel to line M in the figure above and intersects the negative direction of Y-axis?
Indicate all such equations.
[Note: Select one or more answer choices]
\includegraphics[width=0.7\linewidth]{04.png}
Indicate all such equations.
[Note: Select one or more answer choices]
\includegraphics[width=0.7\linewidth]{04.png}
Show Hint
For equations in the standard form \(Ax + By = C\), the slope is always \(-\frac{A}{B}\) and the y-intercept is \(\frac{C}{B}\). To find parallel lines to \(4y-3x=\text{const}\) (or \( -3x+4y=\text{const} \)), you just need to find equations of the form \(-3x+4y=C'\) or \(3x-4y=C''\). The slope will be \(-(-3)/4 = 3/4\). Then check the y-intercept condition. A negative y-intercept \(\frac{C'}{4}\) means \(C'\) must be negative.
Updated On: Oct 3, 2025
Hide Solution
Verified By Collegedunia
Solution and Explanation
Step 1: Understanding the Concept:
We need to find equations of lines that satisfy two conditions: 1. Parallel to line M: Two lines are parallel if and only if they have the same slope. 2. Intersects the negative direction of Y-axis: This means the y-intercept of the line must be negative. First, we'll find the slope of line M from the given points. Then, we'll find the slope and y-intercept of each equation in the options to see which ones meet the criteria.
Step 2: Key Formula or Approach:
1. Calculate the slope of line M using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). The given points are (-4, 0) and (0, 3). 2. Convert each of the given equations into the slope-intercept form, \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. 3. Compare the slope of each option with the slope of line M. They must be equal. 4. For the equations with the correct slope, check if their y-intercept \(c\) is negative (\(c<0\)).
Step 3: Detailed Explanation:
Find the slope of line M:
Line M passes through the points \((-4, 0)\) and \((0, 3)\). \[ m_M = \frac{3 - 0}{0 - (-4)} = \frac{3}{4} \] So, we are looking for lines with a slope of \(\frac{3}{4}\) and a negative y-intercept.
Analyze the options:
We will convert each equation to the form \(y = mx + c\).
% Option A: 4y+3x=0
\(4y = -3x \implies y = -\frac{3}{4}x\). Slope is \(-\frac{3}{4}\). Not parallel.
% Option B: 4y-3x=-2
\(4y = 3x - 2 \implies y = \frac{3}{4}x - \frac{2}{4} = \frac{3}{4}x - \frac{1}{2}\). Slope is \(m = \frac{3}{4}\) (parallel). Y-intercept is \(c = -\frac{1}{2}\) (negative). This is a correct choice.
% Option C: 4y-3x=4
\(4y = 3x + 4 \implies y = \frac{3}{4}x + 1\). Slope is \(m = \frac{3}{4}\) (parallel). Y-intercept is \(c = 1\) (positive). Incorrect.
% Option D: 4y+3x=-4
\(4y = -3x - 4 \implies y = -\frac{3}{4}x - 1\). Slope is \(-\frac{3}{4}\). Not parallel.
% Option E: 4y-3x=-1
\(4y = 3x - 1 \implies y = \frac{3}{4}x - \frac{1}{4}\). Slope is \(m = \frac{3}{4}\) (parallel). Y-intercept is \(c = -\frac{1}{4}\) (negative). This is a correct choice.
% Option F: 4y-3x=0
\(4y = 3x \implies y = \frac{3}{4}x\). Slope is \(m = \frac{3}{4}\) (parallel). Y-intercept is \(c = 0\) (not negative). Incorrect.
Step 4: Final Answer:
The equations that represent a line parallel to M and intersect the negative y-axis are 4y-3x=-2 and 4y-3x=-1.
We need to find equations of lines that satisfy two conditions: 1. Parallel to line M: Two lines are parallel if and only if they have the same slope. 2. Intersects the negative direction of Y-axis: This means the y-intercept of the line must be negative. First, we'll find the slope of line M from the given points. Then, we'll find the slope and y-intercept of each equation in the options to see which ones meet the criteria.
Step 2: Key Formula or Approach:
1. Calculate the slope of line M using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). The given points are (-4, 0) and (0, 3). 2. Convert each of the given equations into the slope-intercept form, \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. 3. Compare the slope of each option with the slope of line M. They must be equal. 4. For the equations with the correct slope, check if their y-intercept \(c\) is negative (\(c<0\)).
Step 3: Detailed Explanation:
Find the slope of line M:
Line M passes through the points \((-4, 0)\) and \((0, 3)\). \[ m_M = \frac{3 - 0}{0 - (-4)} = \frac{3}{4} \] So, we are looking for lines with a slope of \(\frac{3}{4}\) and a negative y-intercept.
Analyze the options:
We will convert each equation to the form \(y = mx + c\).
% Option A: 4y+3x=0
\(4y = -3x \implies y = -\frac{3}{4}x\). Slope is \(-\frac{3}{4}\). Not parallel.
% Option B: 4y-3x=-2
\(4y = 3x - 2 \implies y = \frac{3}{4}x - \frac{2}{4} = \frac{3}{4}x - \frac{1}{2}\). Slope is \(m = \frac{3}{4}\) (parallel). Y-intercept is \(c = -\frac{1}{2}\) (negative). This is a correct choice.
% Option C: 4y-3x=4
\(4y = 3x + 4 \implies y = \frac{3}{4}x + 1\). Slope is \(m = \frac{3}{4}\) (parallel). Y-intercept is \(c = 1\) (positive). Incorrect.
% Option D: 4y+3x=-4
\(4y = -3x - 4 \implies y = -\frac{3}{4}x - 1\). Slope is \(-\frac{3}{4}\). Not parallel.
% Option E: 4y-3x=-1
\(4y = 3x - 1 \implies y = \frac{3}{4}x - \frac{1}{4}\). Slope is \(m = \frac{3}{4}\) (parallel). Y-intercept is \(c = -\frac{1}{4}\) (negative). This is a correct choice.
% Option F: 4y-3x=0
\(4y = 3x \implies y = \frac{3}{4}x\). Slope is \(m = \frac{3}{4}\) (parallel). Y-intercept is \(c = 0\) (not negative). Incorrect.
Step 4: Final Answer:
The equations that represent a line parallel to M and intersect the negative y-axis are 4y-3x=-2 and 4y-3x=-1.
Was this answer helpful?
0
0
Top Questions on Geometry
- A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting, and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts required to construct the wire frame to form the cube?
In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and the line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?
Note: The figure shown is representative
\( AB \) is a diameter of the circle. Compare:
Quantity A: The length of \( AB \)
Quantity B: The average (arithmetic mean) of the lengths of \( AC \) and \( AD \).
O is the center of the circle above.
- In the figure above, \( O \) and \( P \) are the centers of the two circles. If each circle has radius \( r \), what is the area of the shaded region?
View More Questions
Questions Asked in GRE exam
- Should we really care for the greatest actors of the past could we have them before us? Should we find them too different from our accent of thought, of feeling, of speech, in a thousand minute particulars which are of the essence of all three? Dr. Doran's long and interesting records of the triumphs of Garrick, and other less familiar, but in their day hardly less astonishing, players, do not relieve one of the doubt. Garrick himself, as sometimes happens with people who have been the subject of much anecdote and other conversation, here as elsewhere, bears no very distinct figure. One hardly sees the wood for the trees. On the other hand, the account of Betterton, "perhaps the greatest of English actors," is delightfully fresh. That intimate friend of Dryden, Tillatson, Pope, who executed a copy of the actor's portrait by Kneller which is still extant, was worthy of their friendship; his career brings out the best elements in stage life. The stage in these volumes presents itself indeed not merely as a mirror of life, but as an illustration of the utmost intensity of life, in the fortunes and characters of the players. Ups and downs, generosity, dark fates, the most delicate goodness, have nowhere been more prominent than in the private existence of those devoted to the public mimicry of men and women. Contact with the stage, almost throughout its history, presents itself as a kind of touchstone, to bring out the bizarrerie, the theatrical tricks and contrasts, of the actual world.
- GRE - 2025
- Reading Comprehension
- “Reviving the practice of using elements of popular music in classical composition, an approach that had been in hibernation in the United States during the 1960s, composer Philip Glass (born 1937) embraced the ethos of popular music in his compositions. Glass based two symphonies on music by rock musicians David Bowie and Brian Eno, but the symphonies' sound is distinctively his. Popular elements do not appear out of place in Glass's classical music, which from its early days has shared certain harmonies and rhythms with rock music. Yet this use of popular elements has not made Glass a composer of popular music. His music is not a version of popular music packaged to attract classical listeners; it is high art for listeners steeped in rock rather than the classics.
- GRE - 2025
- Reading Comprehension
- Called by some the “island that time forgot,” Madagascar is home to a vast array of unique, exotic creatures. One such animal is the aye-aye. First described by western science in 1782, it was initially categorized as a member of the order Rodentia. Further research then revealed that it was more closely related to the lemur, a member of the primate order. Since the aye-aye is so different from its fellow primates, however, it was given its own family: Daubentoniidae. The aye-aye has been listed as an endangered species and, as a result, the government of Madagascar has designated an island off the northeastern coast of Madagascar as a protected reserve for aye-ayes and other wildlife.
Long before Western science became enthralled with this nocturnal denizen of Madagascar’s jungles, the aye-aye had its own reputation with the local people. The aye aye is perhaps best known for its large, round eyes and long, extremely thin middle finger. These adaptations are quite sensible, allowing the aye-aye to see well at night and retrieve grubs, which are one of its primary food sources, from deep within hollow branches. However, the aye-aye’s striking appearance may end up causing its extinction. The people of Madagascar believe that the aye-aye is a type of spirit animal, and that its appearance is an omen of death. Whenever one is sighted, it is immediately killed. When combined with the loss of large swaths of jungle habitat, this practice may result in the loss of a superb .- GRE - 2025
- Reading Comprehension
- It has been suggested that long-term prisoners, on release from jail, be given a reasonable state pension to reduce the likelihood of their resorting to crime. Most people instinctively reject the suggestion as they feel it would be like rewarding criminal activity. The supporters of the prisoners' pension scheme have criticized those who reject this possibility, by claiming that for the critics...
Which of the following is the most logical completion of the sentence above?- GRE - 2025
- Critical Reasoning
- Jean Valjean, the protagonist in Victor Hugo’s novel Les Misérables, spends the bulk of his lengthy _______ plotting a very clever set of revenge schemes, which he executes upon his escape.
- GRE - 2025
- Vocabulary
View More Questions