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MHTCET 22 Apr 2025 Shift 1 Paper

Question : 109 of 150

Marks: +1, -0

If the planes

\overline{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-\lambda \hat{\mathrm{j}}+\hat{\mathrm{k}})=3

\overline{\mathrm{r}} \cdot(4 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\mu \hat{\mathrm{k}})=5

\lambda+\mu=

Solution:

Concept:

For two planes to be parallel, their normal vectors must be proportional.

Explanation:

The given planes have normal vectors

\overline{\mathrm{n}}_1 = 2\hat{\mathrm{i}} - \lambda\hat{\mathrm{j}} + \hat{\mathrm{k}}

and

\overline{\mathrm{n}}_2 = 4\hat{\mathrm{i}} - \hat{\mathrm{j}} + \mu\hat{\mathrm{k}}

Since the planes are parallel, the ratios of corresponding coefficients must be equal:

\frac{2}{4} = \frac{-\lambda}{-1} = \frac{1}{\mu}

Simplifying the first ratio gives

\frac{2}{4} = \frac{1}{2}

From

\frac{-\lambda}{-1} = \frac{1}{2}

, we get

\lambda = \frac{1}{2}

From

\frac{1}{\mu} = \frac{1}{2}

, we get

\mu = 2

Therefore,

\lambda + \mu = \frac{1}{2} + 2 = \frac{5}{2}

Answer:

\lambda + \mu = \frac{5}{2}

Correct option: C.

\frac{5}{2}

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