Verified Solution • 2025–26 Session moderate

QJEEM22Mab662: JEE Main 2022 Mathematics - Class 12 JEEM22Mab662 Solution: Let a line having direction ratios, 1, − 4, 2 intersect the…

Let a line having direction ratios, 1, −4, 2 intersect the lines x−73=y−1−1=z+21 and x2=y−73=z1 at the points A and B. Then (AB)2 is equal to ___________.

Step-by-Step Verified Solution

Let A(3λ+7,−λ+1,λ−2)A(3 \lambda+7,-\lambda+1, \lambda-2) and B(2μ,3μ+7,μ)B(2 \mu, 3 \mu+7, \mu)

So, DR's of AB∝3λ−2μ+7,−(λ+3μ+6),λ−μA B \propto 3 \lambda-2 \mu+7,-(\lambda+3 \mu+6), \lambda-\mu −2-2

Clearly 3λ−2μ+71=λ+3μ+64=λ−μ−22\frac{3 \lambda-2 \mu+7}{1}=\frac{\lambda+3 \mu+6}{4}=\frac{\lambda-\mu-2}{2}

⇒5λ−3μ=−16\Rightarrow 5 \lambda-3 \mu=-16

And λ−5μ=10\lambda-5 \mu=10

From (i) and (ii) we get λ=−5,μ=−3\lambda=-5, \mu=-3

So, AA is (−8,6,−7)(-8,6,-7) and BB is (−6,−2,−3)(-6,-2,-3)

AB=4+64+16⇒(AB)2=84

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