A- B- C and D are four points such that AB -m2 -6 -2 k- BC -2 - and CD - n -6 -15 -3 k- If - -D- intersects AB atsome point E- then which of the following options is-are correct-A- m -1-2B- n -1-3C- m - nD- m -1-4

The correct options are A m ≥1/2 B n ≥1/3 Let EB = p (AB) and (CE) = q (CD) Then 0 lt; p, q ≤ 1 Since EB + BC + CE = 0 ⇒ pm (2 î 6j + 2 k̂) + (î 2j) + ...

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Byju's Answer

Standard XII

Physics

Thermal Resistance

A, B, C and D...

Question

A, B, C and D are four points such that $- - \to A B = m (2^i - 6^j + 2^k), - - \to B C = (^i - 2^j) and - - \to C D = n (- 6^i + 15^j - 3^k) .$ If $^C D$ intersects $- - \to A B$ at some point E, then which of the following option(s) is/are correct?

$m \geq \frac{1}{2}$

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$n \geq \frac{1}{3}$

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$m = n$

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$m \geq \frac{1}{4}$

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Solution

The correct options are
A
$m \geq \frac{1}{2}$

B
$n \geq \frac{1}{3}$

Let $- - \to E B = p (- - \to A B) a n d (- - \to C E) = q (- - \to C D)$

Then $0 < p, q \leq 1$

Since $- - \to E B + - - \to B C + - - \to C E = 0$

$\Rightarrow p m (2^i - 6 j + 2^k) + (^i - 2 j) + q n (- 6^i + 15 j - 3^k) = 0$

$\Rightarrow (2 p m + 1 - 6 q n)^i + (- 6 p m - 2 + 15 q n)^j + (2 p m - 3 q n)^k = 0$

$\Rightarrow 2 p m - 6 q n + 1 = 0, - 6 p m - 2 + 15 q n = 0, 2 p m - 3 q n = 0$

$p = \frac{1}{2 m}, q = \frac{1}{3 n}$

$∴ 0 < \frac{1}{2 m} \leq 1 a n d 0 < \frac{1}{3 n} \leq 1$

$\Rightarrow m \geq \frac{1}{2}, n \geq \frac{1}{3}$

Suggest Corrections

Similar questions

A, B, C

and

D

are four points such that

A B = m (2 i - 6 j + 2 k), B C = i - 2 j

and

C D = n (- 6 i + 15 j - 3 k)

. If

A B

and

C D

intersect at some point

E

, then

Q. Let

A (\to a),

B (\to b),

C (\to c)

and

D (\to d)

be four points such that

\to a = - 2^i + 4^j + 3^k,

\to b = 2^i - 8^j,

\to c =^i - 3^j + 5^k,

\to d = 4^i +^j - 7^k .

m

is the shortest distance between the lines

A B

and

C D,

then which of the following is (are) CORRECT?

Q. Th position vector of four points

A, B, C

and

D

are

^i +^j +^k, 2^i + 5^j, 3^i + 2^j - 3^k,^i - 6^j -^k

.
Then select the correct relation.

If^i +^j +^k,

2^i + 5^j,

3^i + 2^j - 3^k,

^i - 6^j -^k

respectively are the position vectors of points

A,

B,

C

and

D,

then find the angle between the straight lines

A B

and

C D .

Find whether

- - \to A B

and

- - \to C D

are collinear or not.

If^i +^j +^k,

2^i + 5^j,

3^i + 2^j - 3^k,

^i - 6^j -^k

respectively are the position vectors of points

A,

B,

C

and

D,

then find the angle between the straight lines

A B

and

C D .

Find whether

- - \to A B

and

- - \to C D

are collinear or not.

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A, B, C and D are four points such that −−→AB=m(2^i−6^j+2^k), −−→BC=(^i−2^j) and −−→CD=n(−6^i+15^j−3^k). If ^CD intersects −−→AB at some point E, then which of the following option(s) is/are correct?

A, B, C and D are four points such that $- - \to A B = m (2^i - 6^j + 2^k), - - \to B C = (^i - 2^j) and - - \to C D = n (- 6^i + 15^j - 3^k) .$ If $^C D$ intersects $- - \to A B$ at some point E, then which of the following option(s) is/are correct?