Question

Let \( \alpha, \beta \) and \( \gamma \) be the angles made by a straight line with the x-axis, y-axis and z-axis respectively. If \( \cos\alpha + \cos\beta + \cos\gamma = \frac{5}{3} \), then the value of \( \cos\alpha \cos\beta + \cos\beta \cos\gamma + \cos\gamma \cos\alpha \) is equal to

• A. \( \frac{11}{3} \)
• B. \( \frac{8}{9} \)
• C. \( \frac{11}{9} \)
• D. \( \frac{7}{3} \)
• E. \( \frac{7}{9} \)

Hint:

Always use \( l^2+m^2+n^2=1 \) with \( (l+m+n)^2 \) identity for such problems.

Answer 1

The Correct Option is B

Note: There appears to be a typo in the OCR of the question. Based on the correct answer and standard problems of this type, the given condition should be \(\cos \alpha + \cos \beta + \cos \gamma = \frac{5}{3}\). The provided image confirms this.
Step 1: Understanding the Concept:
The values \(\cos\alpha, \cos\beta, \cos\gamma\) are the direction cosines of a line, often denoted as l, m, and n. They are related by a fundamental identity. This problem combines this geometric identity with an algebraic expansion.
Step 2: Key Formula or Approach:
1. The fundamental identity for direction cosines: \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\). 2. The algebraic identity for the square of a trinomial: \((a+b+c)^2 = a^2+b^2+c^2 + 2(ab+bc+ca)\).
Step 3: Detailed Explanation:
Let \(l = \cos\alpha, m = \cos\beta, n = \cos\gamma\). We are given: \[ l + m + n = \frac{5}{3} \] We need to find the value of \(lm + mn + nl\). Let's square the given equation: \[ (l + m + n)^2 = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \] Using the algebraic identity, we expand the left side: \[ l^2 + m^2 + n^2 + 2(lm + mn + nl) = \frac{25}{9} \] Now, substitute the identity for direction cosines, \(l^2 + m^2 + n^2 = 1\): \[ 1 + 2(lm + mn + nl) = \frac{25}{9} \] Now, solve for the expression \(lm + mn + nl\): \[ 2(lm + mn + nl) = \frac{25}{9} - 1 \] \[ 2(lm + mn + nl) = \frac{25 - 9}{9} = \frac{16}{9} \] \[ lm + mn + nl = \frac{16/9}{2} = \frac{8}{9} \] Substituting back the cosine terms: \[ \cos\alpha\cos\beta + \cos\beta\cos\gamma + \cos\gamma\cos\alpha = \frac{8}{9} \] Step 4: Final Answer:
The value of the expression is \(\frac{8}{9}\).