In a - PQR - let a-QR-b-RP andc- PQ- If - a-12- - b-4-3 and b- c-24- then which of the following is-are true-

Using vectors summation method, a=b+c Squaring both sides, we have: nbsp; |a|^2=|c|^2+|b|^2+2 c·b ⇒ |c|^2=12^2 (4 √(3))^2 2 × 24 ⇒c=4 √(3) ∴|c|^22 |a|=12 Now ...

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Standard X

Mathematics

Section Formula

In a Δ PQR , ...

Question

In a $Δ P Q R,$ let $\to a = - - \to Q R, \to b = - - \to R P$ and $\to c = - - \to P Q$ . If $| \to a | = 12, | \to b | = 4 \sqrt{3}$ and $\to b . \to c = 24,$ then which of the following is/are true:

\frac{{∣ ∣ \to c ∣ ∣}^{2}}{2} - ∣ ∣ \to a ∣ ∣ = 12

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\frac{{∣ ∣ \to c ∣ ∣}^{2}}{2} + ∣ ∣ \to a ∣ ∣ = 30

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∣ ∣ ∣ \to a \times \to b + \to c \times \to a ∣ ∣ ∣ = 48 \sqrt{3}

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\to a . \to b = - 72

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Solution

The correct option is D $\to a . \to b = - 72$
Using vectors summation method, $- \to a = \to b + \to c$
Squaring both sides, we have:
$| \to a |^{2} = | \to c |^{2} + | \to b |^{2} + 2 \to c \cdot \to b$
$\Rightarrow | \to c |^{2} = 12^{2} - (4 \sqrt{3})^{2} - 2 \times 24$
$\Rightarrow \to c = 4 \sqrt{3}$
$∴ \frac{| \to c |^{2}}{2} - | \to a | = 12$
Now, we also have:
$- \to c = \to a + \to b$
Squaring both sides, we have:
$\Rightarrow | \to c |^{2} = 12^{2} + (4 \sqrt{3})^{2} + 2 \to a \cdot \to b$
$\Rightarrow 144 + 48 + 2 \to a . \to b = 48 \Rightarrow \to a . \to b = - 72$
$∵ \to a + \to b + \to c = \to 0 \Rightarrow \to a \times \to b + \to a \times \to c = \to a \times \to 0$
$\Rightarrow ∣ ∣ ∣ \to a \times \to b + \to c \times \to a ∣ ∣ ∣ = 2 ∣ ∣ ∣ \to a \times \to b ∣ ∣ ∣$
$= 2 \sqrt{| \to a |^{2} | \to b |^{2} - {(\to a . \to b)}^{2}} = 2 \sqrt{(144) (48) - {(72)}^{2}} = 48 \sqrt{3}$

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Similar questions

Let $Δ P Q R$ be a triangle. Let a = QR, b = RP and c = PQ. If $| a | = 12, | b | = 4 \sqrt{3}$ and $b . c = 24,$ then which of the following is/are true ?