Question

If the plane \(\overset{̄}{r} = (λ+μ)\hat{i}+(2+μ)\hat{j}+(3λ+2μ)\hat{k}\), where \(λ\) and \(μ\) are parameters, intersects coordinate axes at points \((a,0,0)\), \((0,b,0)\) and \((0,0,c)\) then \(a+b+c =\)...

• A. \(\frac{7}{2}\)
• B. \(5\)
• C. \(\frac{8}{3}\)
• D. \(\frac{10}{3}\)

Hint:

Eliminate the parameters to get the Cartesian equation.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The plane is \(x = \lambda + \mu\), \(y = 2 + \mu\), \(z = 3\lambda + 2\mu\). Eliminating \(\lambda\) and \(\mu\) gives a linear equation in \(x, y, z\).

Step 2: Key Formula or Approach:
From \(y\): \(\mu = y - 2\). From \(x\): \(\lambda = x - \mu = x - y + 2\).

Step 3: Detailed Explanation:
\(z = 3(x - y + 2) + 2(y - 2) = 3x - 3y + 6 + 2y - 4 = 3x - y + 2\).
So the plane is \(3x - y - z = -2\), i.e. \(\frac{x}{-2/3} + \frac{y}{2} + \frac{z}{2} = 1\).
Intercepts: \(a = -\frac23\), \(b = 2\), \(c = 2\).
\[ a + b + c = -\frac23 + 4 = \frac{10}{3} \]

Final Answer:
The sum is \(\frac{10}{3}\), option (D). \[ \boxed{\frac{10}{3}} \]