If the lines r -2 a 1 -3 a 2 - b 1 - and r - a 2 - b 2 - are coplanar and a 1 - b 1 - b 2 - ka 2 - b 1 - b 2 - then k-

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Condition for a Conic to Be a Pair of Straight Lines

If the lines ...

Question

If the lines $\vec{r} = 2 \vec{a_{1}} - 3 \vec{a_{2}} + λ \vec{b_{1}} a n d \vec{r} = \vec{a_{2}} + μ \vec{b_{2}}$ are coplanar and $[\vec{a_{1}} \vec{b_{1}} \vec{b_{2}}] = k [\vec{a_{2}} \vec{b_{1}} \vec{b_{2}}]$ , then k =__________.

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\vec{r} = \vec{a_{1}} + λ \vec{b_{1}} a n d \vec{r} = 2 \vec{a_{2}} + λ \vec{b_{2}}

\frac{[\vec{a_{1}} \vec{b_{1}} \vec{b_{2}}]}{[\vec{a_{2}} \vec{b_{1}} \vec{b_{2}}]}

(a_{1}, b_{1})

(a_{2}, b_{2})

(a_{1} + a_{2}, b_{1} + b_{2})

a_{1} b_{2} = a_{2} b_{1}

If two lines L_{1} = a_{1} + {λb}_{1} and L_{2} = a_{2} + {λb}_{2} are such that b_{1} is parallel to b_{2} and a_{1} \neq a_{2}, then L_{1} and L_{2} will be

In the given two lines L_{1} = a_{1} + {λb}_{1} and L_{2} = a_{2} + {λb}_{2}, if b_{1} = b_{2} then the lines will be definitely parallel .

Parallel lines are obtained for the pair of equations $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ if

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