JEE Maths›Straight Lines›Visual Solution
Visual SolutionPYQ 2024 · Jan Shift 2Tricky
Question
If the sum of squares of all real values of α\alpha, for which the lines 2x−y+3=02x - y + 3 = 0, 6x+3y+1=06x + 3y + 1 = 0 and αx+2y−2=0\alpha x + 2y - 2 = 0 do not form a triangle, is pp, then the greatest integer less than or equal to pp is:
Solution Path
Lines don't form a triangle when parallel or concurrent: α=−4,4,45\alpha = -4, 4, \frac{4}{5}. Sum of squares =32.64= 32.64, so ⌊p⌋=32\lfloor p \rfloor = 32.
01Question Setup
1/4
Three lines L1,L2,L3L_1, L_2, L_3 with parameter α\alpha. Find ⌊p⌋\lfloor p \rfloor where pp is sum of squares of α\alpha values for which lines don't form a triangle.
⌊p⌋=  ?\lfloor p \rfloor = \;?
02When No Triangle?
2/4
Parallel cases: L3∥L1⇒α=−4L_3 \parallel L_1 \Rightarrow \alpha = -4, L3∥L2⇒α=4L_3 \parallel L_2 \Rightarrow \alpha = 4. Concurrent: determinant =0⇒α=45= 0 \Rightarrow \alpha = \frac{4}{5}.
α=−4,4,45\alpha = -4, 4, \tfrac{4}{5}
03Sum of SquaresKEY INSIGHT
3/4
p=16+16+1625=32.64p = 16 + 16 + \frac{16}{25} = 32.64. So ⌊p⌋=32\lfloor p \rfloor = 32.
⌊p⌋=32\lfloor p \rfloor = 32
04Final Answer
4/4
⌊p⌋=32\lfloor p \rfloor = 32.
32\boxed{32}
Concepts from this question1 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
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