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JEE Main 2026 · 4 Apr · Shift 1, Mathematics question 14

A line with direction ratios 1,−1,21,-1,2 intersects the lines x2=y3=z+13\dfrac x2=\dfrac y3=\dfrac{z+1}3 and x+1−1=y−21=z4\dfrac{x+1}{-1}=\dfrac{y-2}1=\dfrac z4 at the points P and Q, respectively. If the length of the line segment PQ is α\alpha, then 225α2225\alpha^2 is equal to:

Choose one option
Answer and worked solution

Official answer: B

General points. P=(2s,3s,3s−1)P=(2s,3s,3s-1) and Q=(−1−t, 2+t, 4t)Q=(-1-t,\,2+t,\,4t), with Q−P=λ(1,−1,2)Q-P=\lambda(1,-1,2).

Solve.

  • Adding the x and y components: 1−5s=01-5s=0, so s=15s=\frac15.
  • Then λ=−t−75\lambda=-t-\frac75, and the z component gives 4t+25=2λ4t+\frac25=2\lambda, so t=−815t=-\frac8{15} and λ=−1315\lambda=-\frac{13}{15}.

Length. α2=λ2⋅6=169⋅6225\alpha^2=\lambda^2\cdot6=\dfrac{169\cdot6}{225}, so 225α2=1014225\alpha^2=1014.

(B)

Topics:Equation of a line in spaceAngles and distances
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