Let two non-collinear unit vectors a and b form an acute angle- A point P moves so that at any time t- the position vector OP- where O is the origin- is given by acos t-bsin t when P is farthest from O- let M be the length of OP and u be unit vector along OP then

The correct option is B u=a+b|a+b| and M= (1+a.b)^12 u=acos t+bsint ⇒|OP|^2=|a|^2cos^2t+|b|^2sin^2t+2.a.b.sintcost =cos^2t+sin^2t+b+2 ...

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Byju's Answer

Standard X

Mathematics

Slope of Line Using Trigonometry

Let two non-c...

Question

Let two non-collinear unit vectors $\to a$ and $\to b$ form an acute angle. A point $P$ moves so that at any time $t$ , the position vector $- - \to O P$ , where $O$ is the origin, is given by $\to a cos t + \to b sin t$ when $P$ is farthest from $O$ , let $M$ be the length of $- - \to O P$ and $\to u$ be unit vector along $- - \to O P$ then

\to u = \frac{\to a - \to b}{∣ ∣ ∣ \to a - \to b ∣ ∣ ∣}

and

M = {(1 + \to a . \to b)}^{\frac{1}{2}}

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\to u = \frac{\to a + \to b}{∣ ∣ ∣ \to a + \to b ∣ ∣ ∣}

and

M = {(1 + \to a . \to b)}^{\frac{1}{2}}

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\to u = \frac{\to a - \to b}{∣ ∣ ∣ \to a - \to b ∣ ∣ ∣}

and

M = {(1 + 2 \to a . \to b)}^{\frac{1}{2}}

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\to u = \frac{\to a + \to b}{∣ ∣ ∣ \to a + \to b ∣ ∣ ∣}

and

M = {(1 + 2 \to a . \to b)}^{\frac{1}{2}}

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Solution

The correct option is B $\to u = \frac{\to a + \to b}{∣ ∣ ∣ \to a + \to b ∣ ∣ ∣}$ and $M = {(1 + \to a . \to b)}^{\frac{1}{2}}$
$\to u = \to a cos t + \to b sin t$
$\Rightarrow {∣ ∣ ∣ - - \to O P ∣ ∣ ∣}^{2} = {∣ ∣ \to a ∣ ∣}^{2} {cos}^{2} t + {∣ ∣ ∣ \to b ∣ ∣ ∣}^{2} {sin}^{2} t + 2. \to a . \to b . sin t cos t$
$= {cos}^{2} t + {sin}^{2} t + \to b + 2. \to a . \to b sin t cos t$
$= 1 + 2. \to a . \to b sin t cos t$
$= 1 + \to a . \to b sin 2 t$
${∣ ∣ ∣ - - \to O P ∣ ∣ ∣}_{m a x} = M = \sqrt{1 + \to a . \to b}$ for $t = \frac{π}{4}$
$∴ \to u = \frac{\to a + \to b}{∣ ∣ ∣ \to a + \to b ∣ ∣ ∣}$

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Let two non-collinear unit vectors $^a and^b$ form an acute angle. A point P moves so that at any time t the position vector
$\to O P$ (where, O is the origin) is given by
$^a cos t +^b sin t$ . When P is farthest from origin O, let M be the length of $\to O P$ and $^u$ be the unit vector along $\to O P$ . Then,