SATHEE: 3D Geometry 3 Question 6

3D Geometry 3 Question 6

6. If the plane $2 x - y + 2 z + 3 = 0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4 x - 2 y + 4 z + λ = 0$ and $2 x - y + 2 z + μ = 0$ , respectively, then the maximum value of $λ + μ$ is equal to

(2019 Main, 10 April II)

(a) 13

(b) 15

(d) 9

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Answer:

(a)

Solution:

Equation of given planes are

$2 x - y + 2 z + 3 = 0$

$4 x - 2 y + 4 z + λ = 0$

and

$2 x - y + 2 z + μ = 0$

$∵$ Distance between two parallel planes

$\begin{aligned} a x + b y + c z + d_{1} = 0 \\ and a x + b y + c z + d_{2} = 0 is \end{aligned}$

distance $= \frac{| d_{1} - d_{2} |}{\sqrt{a^{2} + b^{2} + c^{2}}}$

$∴$ Distance between planes (i) and (ii) is

$\frac{| λ - 2 (3) |}{\sqrt{16 + 4 + 16}} = \frac{1}{3}$

[given]

$\Rightarrow | λ - 6 | = 2 \Rightarrow λ - 6 = \pm 2 \Rightarrow λ = 8$ or 4 and distance between planes (i) and (iii) is

$\begin{array}{cc} \frac{| μ - 3 |}{\sqrt{4 + 1 + 4}} = \frac{2}{3} \\ \Rightarrow & | μ - 3 | = 2 \\ \Rightarrow & μ - 3 = \pm 2 \Rightarrow μ = 5 or 1 \end{array}$

So, maximum value of $(λ + μ)$ at $λ = 8$ and $μ = 5$ and it is equal to 13.

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3D Geometry 3 Question 6

6. If the plane $2 x - y + 2 z + 3 = 0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4 x - 2 y + 4 z + λ = 0$ and $2 x - y + 2 z + μ = 0$ , respectively, then the maximum value of $λ + μ$ is equal to

Answer:

Solution:

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3D Geometry 3 Question 6

6. If the plane 2x−y+2z+3=0 has the distances 13 and 23 units from the planes 4x−2y+4z+λ=0 and 2x−y+2z+μ=0, respectively, then the maximum value of λ+μ is equal to

Answer:

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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6. If the plane $2 x - y + 2 z + 3 = 0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4 x - 2 y + 4 z + λ = 0$ and $2 x - y + 2 z + μ = 0$ , respectively, then the maximum value of $λ + μ$ is equal to