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Visual SolutionPYQ 2024 · Jan Shift 1Standard
Question
The portion of the line 4x+5y=204x + 5y = 20 in the first quadrant is trisected by the lines L1L_1 and L2L_2 passing through the origin. The tangent of the angle between the lines L1L_1 and L2L_2 is:
(A)85\frac{8}{5}
(B)2541\frac{25}{41}
(C)25\frac{2}{5}
(D)3041\frac{30}{41}
Solution Path
Find intercepts P(5,0)P(5,0), Q(0,4)Q(0,4). Trisection points A(10/3,4/3)A(10/3, 4/3) and B(5/3,8/3)B(5/3, 8/3) give slopes m1=2/5m_1=2/5, m2=8/5m_2=8/5. Apply tan⁡θ=∣m2−m1∣/(1+m1m2)=30/41\tan\theta = |m_2-m_1|/(1+m_1 m_2) = 30/41.
01Question Setup
1/4
Line 4x + 5y = 20 in Q1 is trisected by L₁, L₂ through the origin. Find tan θ between L₁ and L₂.
Find tan⁡θ\tan\theta between L1L_1 and L2L_2
02Find Trisection Points
2/4
Intercepts P(5,0) and Q(0,4). Trisection gives A(10/3, 4/3) and B(5/3, 8/3).
A=(103,43),  B=(53,83)A = \left(\dfrac{10}{3}, \dfrac{4}{3}\right), \; B = \left(\dfrac{5}{3}, \dfrac{8}{3}\right)
03Slopes and Angle FormulaKEY INSIGHT
3/4
Lines through origin: m₁ = (4/3)/(10/3) = 2/5, m₂ = (8/3)/(5/3) = 8/5. Apply tan θ = |m₂-m₁|/(1+m₁m₂).
tan⁡θ=6/541/25=3041\tan\theta = \dfrac{6/5}{41/25} = \dfrac{30}{41}
04Final Answer
4/4
The tangent of the angle between the trisecting lines is 30/41.
tan⁡θ=3041\tan\theta = \dfrac{30}{41} - Answer (D)
Concepts from this question2 concepts unlocked
★

Angle Between Two Lines

EASY

Given slopes m₁ and m₂, the tangent of the acute angle between the lines is |m₁-m₂|/(1+m₁m₂).

tan⁡θ=∣m1−m2∣1+m1m2\tan\theta = \frac{|m_1 - m_2|}{1 + m_1 m_2}

Appears directly in 2-3 JEE questions every year. Quick formula application once you have the slopes.

Angle problemsBisector problemsTrisection
Practice (15 Qs) →
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Section Formula (Internal Division)

EASY

Point dividing line segment from P to Q in ratio m:n is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).

(mx2+nx1m+n,  my2+ny1m+n)\left(\frac{mx_2 + nx_1}{m+n},\; \frac{my_2 + ny_1}{m+n}\right)

Trisection, midpoint, and centroid all reduce to this formula. Essential for coordinate geometry.

TrisectionMidpointCentroidInternal division
Practice (11 Qs) →
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