Difficulty: Easy
Correct Answer: x - y = -4
Explanation:
Introduction / Context:
This question tests understanding of the relationship between the angle a line makes with the positive x axis and its slope, as well as the use of point slope form to obtain the equation of a line. Such coordinate geometry problems are standard in aptitude and school level mathematics exams.
Given Data / Assumptions:
Concept / Approach:
The slope m of a line making an angle θ with the positive x axis is given by:
m = tan θ
For θ = 45°, tan 45° = 1, so the slope is 1. We then use the point slope form:
y - y1 = m(x - x1)
where (x1, y1) is a known point on the line. After that, we rearrange into one of the given standard forms.
Step-by-Step Solution:
Angle with x axis is 45°, so slope m = tan 45° = 1
Point on the line is (0, 4)
Use point slope form: y - 4 = 1(x - 0)
Simplify: y - 4 = x
Rearrange to standard form: x - y + 4 = 0
This can be written as x - y = -4
Verification / Alternative check:
Check whether the point (0, 4) satisfies x - y = -4. Substituting x = 0 and y = 4 gives 0 - 4 = -4, which is true. Also, the slope from the equation y = x + 4 (equivalent form) is 1, confirming that the line makes a 45° angle with the positive x axis.
Why Other Options Are Wrong:
Option a, x + y = 4, gives y = 4 - x, whose slope is -1, corresponding to an angle of 135°, not 45°. Option c, x + y = -4, has slope -1 and does not pass through (0, 4). Option d, x - y = 4, gives y = x - 4, which has slope 1 but passes through (0, -4), not (0, 4). Option e, y = 4x, passes through the origin (0, 0) and has slope 4, so it is not correct either.
Common Pitfalls:
Some students mix up the sign of the slope when converting from standard form to slope intercept form. Others may assume that any line with slope 1 passes through the origin, forgetting the vertical intercept. Always verify both slope and the given point to ensure the correct equation.
Final Answer:
The equation of the required line is x - y = -4.
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