NDA 2026 Mathematics PYQ — The equation of a straight line which passes through the origin a… | Mathem Solvex | Mathem Solvex
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NDA 2026 — Mathematics PYQ
NDA | Mathematics | 2026
The equation of a straight line which passes through the origin and makes an angle α (0∘≤α≤90∘) with the line L:x+3y+33=0 is x−3y=0.
What is the angle made by the line L with positive direction of y-axis ?
एक सरल रेखा, जो मूलबिंदु से गुजरती है और रेखा L:x+3y+33=0 के साथ (0∘≤α≤90∘)कोण बनाती है; का समीकरण x−3y=0 है।
रेखा L, y-अक्ष की धनात्मक दिशा के साथ कौनसा कोण बनाती है?
Choose the correct answer:
A.
30∘
B.
45∘
C.
60∘
(Correct Answer)
D.
90∘
Correct Answer:
60∘
Explanation
To determine the angle that the line L makes with the positive direction of the y-axis, we proceed as follows:
Step 1: Find the slope (m) of the line L
The given equation is x+3y+33=0. We convert this to the slope-intercept form (y=mx+c):
3y=−x−33
y=−31x−3
Comparing this to the standard form y=mx+c, the slope of the line is m=−31.
Step 2: Relate the slope to the angle of inclination
The angle of inclination (θ) with the positive x-axis is given by tanθ=m=−31.
Since tanθ is negative, the angle is in the second quadrant:
θ=180∘−30∘=150∘
Step 3: Calculate the angle with the y-axis
The angle ϕ that a line makes with the positive y-axis is the complement of the angle it makes with the x-axis (measured relative to the 90∘ position). The formula relating the slope to the angle with the y-axis is:
tanϕ=m1
Substituting the value of m:
tanϕ=−1/31
tanϕ=∣−3∣
tanϕ=3
Therefore:
ϕ=tan−1(3)=60∘
Correct Option: (c) 60∘
Explanation
To determine the angle that the line L makes with the positive direction of the y-axis, we proceed as follows:
Step 1: Find the slope (m) of the line L
The given equation is x+3y+33=0. We convert this to the slope-intercept form (y=mx+c):
3y=−x−33
y=−31x−3
Comparing this to the standard form y=mx+c, the slope of the line is m=−31.
Step 2: Relate the slope to the angle of inclination
The angle of inclination (θ) with the positive x-axis is given by tanθ=m=−31.
Since tanθ is negative, the angle is in the second quadrant:
θ=180∘−30∘=150∘
Step 3: Calculate the angle with the y-axis
The angle ϕ that a line makes with the positive y-axis is the complement of the angle it makes with the x-axis (measured relative to the 90∘ position). The formula relating the slope to the angle with the y-axis is:
