JEE Mains · Maths · STD 12 - 11. three dimension geometry
If the equation of a plane \(P ,\) passing through the intesection of the planes, \(x+4 y-z+7=0\) and \(3 x+y+5 z=8\) is \(ax +b y+6 z=15\) for some \(a, b \in R,\) then the distance of the point \((3,2,-1)\) from the plane \(P\) is
- A \(3\)
- B \(7\)
- C \(21\)
- D \(63\)
Answer & Solution
Correct Answer
(A) \(3\)
Step-by-step Solution
Detailed explanation
\(D _{1}=\left|\begin{array}{ccc}-7 & 4 & -1 \\ 8 & 1 & 5 \\ 15 & b & 6\end{array}\right|=0 \Rightarrow b =-3\) \(D=\left|\begin{array}{ccc}1 & 4 & -1 \\ 3 & 1 & 5 \\ a & b & 6\end{array}\right|=0 \Rightarrow 21 a-8 b-66=0 \ldots\) \(P: 2 x-3 y+6 z=15\) so required distance…
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 number of functions \(f :\{1,2,3,4\} \rightarrow\{ a \in Z :| a | \leq 8\}\) satisfying \(f ( n )+\) \(\frac{1}{ n } f ( n +1)=1, \forall n \in\{1,2,3\}\) isJEE Mains 2023 Hard
- If the square of the shortest distance between the lines \(\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}\) and \(\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5}\) is \(\frac{\mathrm{m}}{\mathrm{n}}\), where \(\mathrm{m}, \mathrm{n}\) are coprime numbers, then \(\mathrm{m}+\mathrm{n}\) is equal to :JEE Mains 2025 Medium
- If \([t]\) denotes the greatest integer \(\leq 1\), then the value of \(\frac{3(e-1)^2}{e} \int \limits_1^2 x^2 e^{[x]+\left[x^3\right]} d x\) is :JEE Mains 2023 Hard
- If \(\lim \limits_{x \rightarrow 0}\left\{\frac{1}{x^{8}}\left(1-\cos \frac{x^{2}}{2}-\cos \frac{x^{2}}{4}+\cos \frac{x^{2}}{2} \cos \frac{x^{2}}{4}\right)\right\}=2^{-k}\) then the value of \(k\) isJEE Mains 2020 Medium
- If \(A =\frac{1}{5 ! 6 ! 7 !}\left[\begin{array}{lll}5 ! & 6 ! & 7 ! \\ 6 ! & 7 ! & 8 ! \\ 7 ! & 8 ! & 9 !\end{array}\right]\), then \(|\operatorname{adj}(\operatorname{adj}(2 A ))|\) is equal to:JEE Mains 2023 Hard
- All the pairs \((x, y)\) that satisfy the inequality \({2^{\sqrt {{{\sin }^2}{\kern 1pt} x - 2\sin {\kern 1pt} x + 5} }}.\frac{1}{{{4^{{{\sin }^2}\,y}}}} \leq 1\) also Satisfy the equationJEE Mains 2019 Hard
More PYQs from JEE Mains
- If \(1 + {x^4} + {x^5} = \sum\limits_{i = 0}^5 {{a_i}\,(1 + {x})^i,} \) for all \(x\) in \(R,\) then \(a_2\) isJEE Mains 2014 Hard
- If the shortest distance between the lines \(\frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1}\) and \(\frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4}\) is \(\frac{44}{\sqrt{30}}\), then the largest possible value of \(|\lambda|\) is equal to ..........JEE Mains 2024 Hard
- The number of real solutions of the equation \(\mathrm{x}|\mathrm{x}+5|+2|\mathrm{x}+7|-2=0\) is ...........JEE Mains 2024 Hard
- The number of all \(3 \times 3\) matrices \(A\), with enteries from the set \(\{-1,0,1\}\) such that the sum of the diagonal elements of \(\mathrm{AA}^{\mathrm{T}}\) is \(3,\) isJEE Mains 2020 Hard
- If \({\cos ^{ - 1}}\,x\, - \,{\cos ^{ - 1}}\,\frac{y}{2}\, = \,\alpha ,\) where \( - {\kern 1pt} 1\, \le \,x\, \le \,1,\,\) \(- {\kern 1pt} 2\, \le \,y\, \le \,2,\) \(x\, \le \,\,\frac{y}{2},\) then for all \(x, y, 4x^2 -4xy\,\,cos\,\alpha + y^2\) is equal toJEE Mains 2019 Hard
- Let \(A =\{ x \in R :| x +1|<2\}\) and \(B=\{x \in R:|x-1| \geq 2\}\). Then which one of the following statements is NOT true ?JEE Mains 2022 Medium