The lines $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}})+\ell(2 \hat{\mathrm{i}}+\hat{\mathrm{k}})$ and $\vec{r}=(2 \hat{i}-\hat{j})+m(\hat{i}+\hat{j}-\hat{k})$

(C) do not intersect for any values of $\ell$ and m

(A) intersect when $\ell=2$ and $\mathrm{m}=\frac{1}{2}$

(B) intersect when $\ell=1$ and $\mathrm{m}=2$

(D) intersect for all values of $\ell$ and m

JEE Main · 2020 · 03 Sep 2020 · Shift 1 · Three Dimensional Geometry

The lines

r = (\hat{i} - \hat{j}) + ℓ (2 \hat{i} + \hat{k})

and

r = (2 \hat{i} - \hat{j}) + m (\hat{i} + \hat{j} - \hat{k})

(A)intersect when $ℓ = 2$ and $m = \frac{1}{2}$
(B)intersect when $ℓ = 1$ and $m = 2$
(C)do not intersect for any values of $ℓ$ and m✓ Correct
(D)intersect for all values of $ℓ$ and m

Answer: (C)

Solution

Equating the two position vectors component by component gives

1 + 2 ℓ = 2 + m

- 1 = m - 1

and

ℓ = - m

. The second equation forces

m = 0

, hence

ℓ = 0

, but these do not satisfy the first, since

1 \neq = 2

. The system is inconsistent, so the lines do not intersect for any values of

ℓ

and

m

JEE Main · 2020 · 03 Sep 2020 · Shift 1 · Single correct option

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