JEE Mains · Maths · STD 12 - 11. three dimension geometry
If the plane \(2x -y + 2z + 3 = 0\) has the distances \(\frac {1}{3}\) and \(\frac {2}{3}\) units from the planes \(4x -2y + 4z + \lambda = 0\) and \(2x -y + 2z + \mu = 0,\) respectively, then the maximum value of \(\lambda + \mu \) us equal to
- A \(15\)
- B \(13\)
- C \(5\)
- D \(9\)
Answer & Solution
Correct Answer
(B) \(13\)
Step-by-step Solution
Detailed explanation
\(4 x-2 y+4 z+6=0\) \(\frac{|\lambda-6|}{\sqrt{16+4+16}}=\left|\frac{\lambda-6}{6}\right|=\frac{1}{3}\) \(|\lambda-6|=2\) \(\lambda=8,4\) \(\frac{|\mu-3|}{\sqrt{4+4+1}}=\frac{2}{3}\) \(|\mu-3|=2\) \(\mu=5,1\) \(\therefore \) Maximum value of \((\mu+\lambda)=13.\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The distance ofthe point \((1,3,-7)\) from the plane passing through the point \(\left( {1, - 1, - 1} \right)\) having normal perpendicular to both the lines \(\frac{{x - 1}}{1} = \frac{{y + 2}}{{ - 2}} = \frac{{z - 4}}{3}\) and \(\frac{{x - 2}}{2} = \frac{{y + 1}}{{ - 1}} = \frac{{z + 7}}{{ - 1}}\) is . . . .JEE Mains 2017 Hard
- Let \(f:(0, \infty) \rightarrow R\) and \(F(x)=\int_0^x t f(t) d t\). If \(F\left(x^2\right)=\) \(x^4+x^5\), then \(\sum_{r=1}^{12} f\left(r^2\right)\) is equal to :JEE Mains 2024 Hard
- The remainder when \(19^{200}+23^{200}\) is divided by \(49\) , is \(.........\).JEE Mains 2023 Hard
- If \( y=y(x) \) satisfies the differential equation \( 16(\sqrt{x+9\sqrt{x}})(4+\sqrt{9+\sqrt{x}})cos~y~dy=(1+2~sin~y)dx, x>0 \) and \( y(256)=\frac{\pi}{2}, y(49)=\alpha \) then \( 2~sin~\alpha \) is equal to:JEE Mains 2026 Easy
- If the domain of the function \(\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _e\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)\) is \((\alpha, \beta]\), then \(3 \alpha+10 \beta\) is equal to :JEE Mains 2024 Hard
- If \(A =\frac{1}{2}\left[\begin{array}{cc}1 & \sqrt{3} \\ -\sqrt{3} & 1\end{array}\right]\), then :JEE Mains 2023 Hard
More PYQs from JEE Mains
- Let \(y = y\left( x \right)\) be the solutions of the differential equation, \(\left( {{x^2} + 1} \right)^2\,\frac{{dy}}{{dx}} + 2x\left( {{x^2} + 1} \right)\,y = 1\) such that \(y\left( 0 \right) = 0\). If \(\sqrt a y\left( 1 \right) = \frac{\pi }{{32}}\), then the value of \(‘a’\) isJEE Mains 2019 Hard
- Let \(A :\{1,2,3,4,5,6,7\}\). Define \(B =\{ T \subseteq A\) : either \(1 \notin T\) or \(2 \in T \}\) and \(C = \{ T \subseteq A : T\) the sum of all the elements of \(T\) is a prime number \(\}\). Then the number of elements in the set \(B \cup C\) is \(\dots\dots\)JEE Mains 2022 Hard
- For a \(3 \times 3\) matrix \(M\), let trace \((M)\) denote the sum of all the diagonal elements of \(M\). Let \(A\) be a \(3 \times 3\) matrix such that \(|A|=\frac{1}{2}\) and trace \((A)=3\). If \(B=\operatorname{adj}(\operatorname{adj}(2 A))\), then the value of \(|B|+\) trace (B) equals :JEE Mains 2025 Medium
- Let \(\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=12 \vec{a}+4 \vec{b}\), and \(\overrightarrow{O C}=\vec{b}\), where \(O\) is the origin. If \(S\) is the parallelogram with adjacent sides \(O A\) and \(O C\), then find the value of \(\frac{\text { area of quadrilateral } O A B C}{\text { area of } S} .\)JEE Mains 2024 Hard
- If \(y = \tan^{-1}\left(\dfrac{3\cos x - 4\sin x}{4\cos x + 3\sin x}\right) + 2\tan^{-1}\left(\dfrac{x}{1+\sqrt{1-x^2}}\right)\), then \(\dfrac{dy}{dx}\) at \(x = \dfrac{\sqrt{3}}{2}\) is equal to:JEE Mains 2026 Medium
- Let \(A=\left[a_{i j}\right]_{2 \times 2}\) where \(a_{i j} \neq 0\) for all \(i, j\) and \(A^2=I\). Let a be the sum of all diagonal elements of \(A\) and \(b =| A |\), then \(3 a ^2+4 b ^2\) is equal toJEE Mains 2023 Hard