Question

need solution for this!!​

Answer 1

Answer:

your answer is d

Step-by-step explanation:

hope it help, mark as Brainlist

Answer 2

★ Concept :-

Here the concept of Distance Formula has been used. We see that the distance between the points has been given. Even the two points are also given. So firstly using the distance formula we can equate the distance and points. Then by simplication we can get the required answer.

Let's do it !!

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★ Formula Used :-

$\;\boxed{\sf{\pink{d\;=\;\bf{\sqrt{(x_{2}\:-\:x_{1})^{2}\:+\:(y_{2}\:-\:y_{1})^{2}}}}}}$

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★ Solution :-

Given,

» Coordinates of first point = (a cos 48°, 0)

» Coordinates of second point = (0, a cos 12°)

» Distance between the two points = d

By distance formula, we know that

$\;\sf{\rightarrow\;\;d\;=\;\bf{\sqrt{(x_{2}\:-\:x_{1})^{2}\:+\:(y_{2}\:-\:y_{1})^{2}}}}$

• Here x₁ = a cos 48°

• Here y₁ = 0

• Here x₂ = 0

• Here y₂ = a cos 12°

By applying these values in the equation, we get

$\;\sf{\Longrightarrow\;\;d\;=\;\bf{\sqrt{(0\:-\:a\cos 48^{\circ})^{2}\:+\:(a\cos 12^{\circ}\:-\:0)^{2}}}}$

$\;\sf{\Longrightarrow\;\;d\;=\;\bf{\sqrt{(-\:a\cos 48^{\circ})^{2}\:+\:(a\cos 12^{\circ})^{2}}}}$

$\;\sf{\Longrightarrow\;\;d\;=\;\bf{\sqrt{a^{2}\cos^{2}48^{\circ}\:+\:a^{2}\cos^{2} 12^{\circ}}}}$

Now taking a² as common from both sides, we get

$\;\sf{\Longrightarrow\;\;d\;=\;\bf{\sqrt{a^{2}(\cos^{2}48^{\circ}\:+\:\cos^{2} 12^{\circ})}}}$

Now squaring both sides, we get

$\;\sf{\Longrightarrow\;\;(d)^{2}\;=\;\bf{\bigg(\sqrt{a^{2}(\cos^{2}48^{\circ}\:+\:\cos^{2} 12^{\circ})}\bigg)^{2}}}$

$\;\sf{\Longrightarrow\;\;d^{2}\;=\;\bf{a^{2}(\cos^{2}48^{\circ}\:+\:\cos^{2} 12^{\circ})}}$

On applying Trigonometric Identity, we get

$\;\sf{\Longrightarrow\;\;d^{2}\;=\;\bf{a^{2}\bigg(\dfrac{1\:+\:\cos 96^{\circ}}{2}\:+\:\dfrac{1\:+\:\cos 24^{\circ}}{2}\bigg)}}$

$\;\sf{\Longrightarrow\;\;2(d^{2})\;=\;\bf{a^{2}(2\:+\:2\cos 60^{\circ}\:\times\:\cos 36^{\circ})}}$

We know that cos 60° = ½ and 2 × ½ = 1

Now subtracting 2a² from both sides we get

$\;\sf{\Longrightarrow\;\;2d^{2}\:-\:2a^{2}\;=\;\bf{a^{2}(2\:+\;1\:\times\:\cos 36^{\circ})}}$

$\;\sf{\Longrightarrow\;\;2(d^{2}\:-\:a^{2})\;=\;\bf{a^{2}\cos 36^{\circ}}}$

$\;\sf{\Longrightarrow\;\;d^{2}\:-\:a^{2}\;=\;\bf{\dfrac{a^{2}\cos 36^{\circ}}{2}}}$

Now we know that, cos 36° = (√5 + 1) / 4

By applying this, we get

$\;\sf{\Longrightarrow\;\;d^{2}\:-\:a^{2}\;=\;\bf{\dfrac{a^{2}\:\times\:\frac{(\sqrt{5}\:+\:1)}{4}}{2}}}$

$\;\bf{\Longrightarrow\;\;\red{d^{2}\:-\:a^{2}\;=\;\bf{\dfrac{a^{2}\:\times\:\sqrt{5}\:+\:1}{8}}}}$

This is the required answer.

So option d) is correct.

$\;\underline{\boxed{\tt{Required\;\:answer\;=\;\bf{\purple{\dfrac{a^{2}(\sqrt{5}\:+\:1)}{8}}}}}}$