You visited us 8 times! Enjoying our articles? Unlock Full Access!

Perpendicular Form of a Straight Line

Let tanα , ta...

Question

Let $\tan α, \tan β$ and $\tan γ; α, β, γ \neq [2 n - 1] π / 2, n \in N$ be the slopes of three-line segments $O A, O B a n d O C$ , respectively, where $O$ is the origin. If the circumcentre of the $∆ A B C$ coincides with the origin and its orthocentre lies on the $y - a x i s,$ then the value of ${[\frac{\cos 3 α + \cos 3 β + \cos 3 γ}{\cos α . \cos β . \cos γ}]}^{2}$ is equal to

Open in App

Solution

Determine the value of ${[\frac{\cos 3 α + \cos 3 β + \cos 3 γ}{\cos α . \cos β . \cos γ}]}^{2}$
The circumcenter and orthocenter both lies on the $y - a x i s$ . Therefore, centroid also lies on the $y - a x i s$
On solving, we get:
$\sum_{} \cos α = 0 \Rightarrow \cos α + \cos β + \cos γ = 0$
We know that if $a + b + c = 0$ , then $a^{3} + b^{3} + c^{3} = 3 a b c$
$\begin{array}{rcl} ∴ \frac{\cos 3 α + \cos 3 β + \cos 3 γ}{\cos α . \cos β . \cos γ} & = & \frac{(4 \cos^{3} α - 3 \cos α) + (4 \cos^{3} β - 3 \cos β) + (4 \cos^{3} γ - 3 \cos γ)}{\cos α . \cos β . \cos γ} \\ = & \frac{4 (\cos^{3} α + \cos^{3} β + \cos^{3} γ) - 3 (\cos α + \cos β + \cos γ)}{\cos α . \cos β . \cos γ} \\ = & \frac{4 (3 \cos α . \cos β . \cos γ) - 3 (0)}{\cos α . \cos β . \cos γ} \\ \Rightarrow & \frac{\cos 3 α + \cos 3 β + \cos 3 γ}{\cos α . \cos β . \cos γ} = 12 \\ \Rightarrow & {(\frac{\cos 3 α + \cos 3 β + \cos 3 γ}{\cos α . \cos β . \cos γ})}^{2} = 144 \end{array}$

Therefore, the value of ${(\begin{array}{rcl} \frac{\cos 3 α + \cos 3 β + \cos 3 γ}{cosα . cosβ . cosγ} \end{array})}^{2} is 144 .$

Suggest Corrections

Similar questions

tan α, tan β, tan γ; α, β, γ \neq \frac{(2 n - 1) π}{2}

n \in N

O A,

O B

O C,

O

△ A B C

y -

{(\frac{cos 3 α + cos 3 β + cos 3 γ}{cos α cos β cos γ})}^{2}

tan α

tan β

tan γ

α + β + γ = π

x (4 cos α / 2 cos β / 2 c o s γ / 2) - y (1 + 4 sin α / 2 sin β / 2 sin γ / 2) = 0

A (x_{1}, x_{1} tan α), B (x_{2}, x_{2} tan β)

C (x_{3}, x_{2} tan γ) .

△ A B C

H (a, b)

\frac{b}{a}

x_{1}, x_{2}, x_{3}

A (p, p tan α), B (q, q tan β), C (r, r tan γ)

O

A B C

H (¯ ¯ ¯ x, ¯ ¯ ¯ y)

\frac{¯ ¯ ¯ x}{¯ ¯ ¯ y} = \frac{cos α + cos β + cos γ}{sin α + sin β + sin γ}

A (x_{1}, x_{1} tan α), B (x_{2}, x_{2} tan β) & C (x_{3}, x_{3} tan γ)

Δ A B C

H (a, b)

\frac{a}{b}

Join BYJU'S Learning Program