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: Find each intercept directly:
On the $x$-axis $y = 0, z = 0$: $\mu = -2$, $z = 0$ gives $3\lambda + 2\mu = 0$, so $\lambda = \frac43$, and $x = \frac43 - 2 = -\frac23$.

Step 2: Other axes:
On the $y$-axis $x = 0, z = 0$: $\lambda = -\mu$, $z = -3\mu + 2\mu = -\mu = 0$, so $\mu = 0$ and $y = 2$.
On the $z$-axis $x = 0, y = 0$: $\mu = -2$, $\lambda = 2$, $z = 6 - 4 = 2$.
Sum $= -\frac23 + 2 + 2 = \frac{10}{3}$.

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