The points A- B and C with position vectors a- b- and c- and respectively lie on a circle centred at origin O- Let G and E be the centroid of - ABC and - ACD respectively where D is midpoint of AB- If OE and CD are mutually perpendicular- then which of the following will be necessarily true-A- - b - a - b - c -B- - c - a - c - b -C- - b - a - c - a -D- - b - a - c - b - b - c -

The correct option is C |b⃗ a⃗|=|c⃗ a⃗| |a⃗|=|b⃗|=|c⃗|=r ...(i) P.V of E: a⃗+a⃗+b⃗/2+c⃗/3 e⃗=3a⃗+b⃗+2c⃗/6 P.V. of G:g⃗=a⃗+b⃗+c⃗/3 Given, OE ⊥ CD ⇒ ( 3 a⃗+b⃗+ ...

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Byju's Answer

Standard XII

Physics

Position Vector

The points A,...

Question

The points A, B and C with position vectors $\to a, \to b a n d \to c$ , and respectively lie on a circle centred at origin O. Let G and E be the centroid of $Δ$ ABC and $Δ$ ACD respectively where D is midpoint of AB.
If OE and CD are mutually perpendicular, then which of the following will be necessarily true?

| \to b - \to a | = | \to b - \to c |

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| \to c - \to a | = | \to c - \to b |

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| \to b - \to a | = | \to c - \to a |

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| \to b - \to a | = | \to c - \to b | = | \to b - \to c |

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Solution

The correct option is C $| \to b - \to a | = | \to c - \to a |$

$| \to a | = | \to b | = | \to c | = r . . . (i)$
P.V of E: $\frac{\to a + \frac{\to a + \to b}{2} + \to c}{3}$
$\to e = \frac{3 \to a + \to b + 2 \to c}{6}$
P.V. of $G : \to g = \frac{\to a + \to b + \to c}{3}$
Given, OE $⊥ C D \Rightarrow (\frac{3 \to a + \to b + 2 \to c}{6}) . (\frac{\to a + \to b}{2} - \to c) = 0$
$\Rightarrow 3 | \to a |^{2} + | \to b |^{2} - 4 | \to c |^{2} + 4 \to a \to b - 4 \to a . \to c = 0$
Hence, $\to O A ⊥ \to B C$
$\Rightarrow Δ A B C m u s t b e i s o s c e l e s w i t h b a s e B C \Rightarrow | ¯ ¯¯¯¯¯¯ ¯ A C | = | ¯ ¯¯¯¯¯¯ ¯ A B | \Rightarrow | \to c - \to a | = | \to b - \to a |$

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