If R-P-Q and R makes an angles - 1 with P and - 2 with Q- Then

The correct options are C If θ _1 lt;θ _2 then | P | gt; | Q | D If θ _1=θ _2 then | P | = | Q | R⃗ = P⃗ + Q⃗ ∠R⃗P⃗ = θ ...

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Byju's Answer

Standard XII

Mathematics

Global Maxima

If R=P+Q an...

Question

If $\to R = \to P + \to Q$ and $\to R$ makes an angles $θ_{1}$ with $\to P$ and $θ_{2}$ with $\to Q$ . Then

θ_{1}

cannot be less than

θ_{2}

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θ_{1} < θ_{2}

then

∣ ∣ ∣ \to P ∣ ∣ ∣ < ∣ ∣ ∣ \to Q ∣ ∣ ∣

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θ_{1} < θ_{2}

then

∣ ∣ ∣ \to P ∣ ∣ ∣ > ∣ ∣ ∣ \to Q ∣ ∣ ∣

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θ_{1} = θ_{2}

then

∣ ∣ ∣ \to P ∣ ∣ ∣ = ∣ ∣ ∣ \to Q ∣ ∣ ∣

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Solution

The correct options are
C If $θ_{1} < θ_{2}$ then $∣ ∣ ∣ \to P ∣ ∣ ∣ > ∣ ∣ ∣ \to Q ∣ ∣ ∣$
D If $θ_{1} = θ_{2}$ then $∣ ∣ ∣ \to P ∣ ∣ ∣ = ∣ ∣ ∣ \to Q ∣ ∣ ∣$
$\to R = \to P + \to Q ∠ \to R \to P = θ_{1} ∠ \to R \to Q = θ_{2} ∠ \to P \to Q = θ cos θ_{1} = \frac{\to R \cdot \to P}{R P} = \frac{(\to P + \to Q) \cdot \to P}{R P} = \frac{P^{2} + \to P \cdot \to Q}{R P} = \frac{P + Q cos θ}{R} . . . . . . (1)$

$cos θ_{2} = \frac{\to R \cdot \to Q}{R Q} = \frac{(\to P + \to Q) \cdot \to Q}{R Q} = \frac{Q^{2} + \to P \cdot \to Q}{R Q} = \frac{Q + P cos θ}{R} . . . . . . (2)$

$cos θ \in [- 1, 1]$
$cos θ$ decreases with $θ$ .
Hence, C and D are correct.

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Similar questions

Q. Given that

P = Q = R

. If

\to P + \to Q = \to R

then the angle between

\to P

and

\to R

θ_{1}

. If

\to P + \to Q + \to R = \to 0

then the angle between

\to P

and

\to R

θ_{2}

. The relation between

θ_{1}

and

θ_{2}

Q. If

\frac{{cos}^{4} θ}{θ_{1}} + \frac{{sin}^{4} θ}{θ_{2}} = \frac{1}{θ_{1} + θ_{2}}

,then

\frac{θ_{2} + θ_{1}}{θ_{1}}

equals

Q. If

\frac{{cos}^{4} x}{θ_{1}} + \frac{{sin}^{4} x}{θ_{2}} = \frac{1}{θ_{1} + θ_{2}},

then

\frac{θ_{2}}{θ_{1}}

equals

Q. If

θ_{1}, θ_{2}

be the inclinations of tangents drawn from the point

P

to the circle

x^{2} + y^{2} = a^{2}

and

cot θ_{1} + cot θ_{2} = k

, then the locus of

P

Q. If two tangents to the parabola

y^{2} = 4 a x

from a point P make angle

θ_{1}

and

θ_{2}

with the axis of the parabola, then the find the locus of P in each of the following cases.

θ_{1} + θ_{2} = α

(a constant}

θ_{1} + θ_{2} = \frac{π}{2}

t a n θ_{1} + t a n θ_{2} = λ

( is constant)

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If →R=→P+→Q and →R makes an angles θ1 with →P and θ2 with →Q. Then

If $\to R = \to P + \to Q$ and $\to R$ makes an angles $θ_{1}$ with $\to P$ and $θ_{2}$ with $\to Q$ . Then