JEE Mains · Maths · STD 12 - 11. three dimension geometry
Let \(P\) be an arbitrary point having sum of the squares of the distance from the planes \(x + y + z =0, l x - nz =0\) and \(x -2 y + z =0\) equal to \(9 .\) If the locus of the point \(P\) is \(x ^{2}+ y ^{2}+ z ^{2}=9,\) then the value of \(l- n\) is equal to ...... .
- A \(0\)
- B \(2\)
- C \(8\)
- D \(10\)
Answer & Solution
Correct Answer
(A) \(0\)
Step-by-step Solution
Detailed explanation
Sol. Let point \(P\) is \((\alpha, \beta, \gamma)\) \(\left(\frac{\alpha+\beta+\gamma}{\sqrt{3}}\right)^{2}+\left(\frac{\ell \alpha- n \gamma}{\sqrt{\ell^{2}+ n ^{2}}}\right)^{2}+\left(\frac{\alpha-2 \beta+\gamma}{\sqrt{6}}\right)^{2}=9\) Locus is…
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 line \(l_1\) passes through the point \((2,6,2)\) and is perpendicular to the plane \(2 x+y-2 z=10\). Then the shortest distance between the line \(l_1\) and the line \(\frac{ x +1}{2}=\frac{ y +4}{-3}=\frac{ z }{2}\) is :JEE Mains 2023 Hard
- The smallest natural number \(n,\) such that the coefficient of \(x\) in the expansion of \({\left( {{x^2}\, + \,\frac{1}{{{x^3}}}} \right)^n}\) is \(^n{C_{23}}\) isJEE Mains 2019 Hard
- If \(\operatorname{Lim}_{x \rightarrow 0}\left(\frac{\tan x}{x}\right)^{\frac{1}{x^2}}=p\), then \(96 \log _e p\) is equal to ______JEE Mains 2025 Medium
- The integral \(16 \int \limits_1^2 \frac{d x}{x^3\left(x^2+2\right)^2}\) is equal toJEE Mains 2023 Hard
- If \(\int \limits_0^\pi \frac{5^{\cos x}\left(1+\cos x \cos 3 x+\cos ^2 x+\cos ^3 x \cos 3 x\right) d x}{1+5^{\cos x}}=\frac{k \pi}{16}\), then \(k\) is equal to \(...........\).JEE Mains 2023 Hard
- A line passing through the point \(\mathrm{A}(-2,0)\), touches the parabola \(P: y^2=x-2\) at the point \(B\) in the first quadrant. The area, of the region bounded by the line AB , parabola P and the x -axis, is :-JEE Mains 2025 Hard
More PYQs from JEE Mains
- Let \(\mathrm{A}=\{-3,-2,-1,0,1,2,3\}\) and R be a relation on \(A\) defined by \(x R y\) if and only if \(2 x-y \in\{0,1\}\). Let \(l\) be the number of elements in R. Let \(m\) and \(n\) be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then \(l+\mathrm{m} \mathrm{n}\) is equal to :-JEE Mains 2025 Easy
- Let \([ x ]\) denote the greatest integer \(\leq x\). Consider the function \(f(x)=\max \left\{x^2, 1+[x]\right\}\). Then the value of the integral \(\int \limits_0^2 f ( x ) dx\) is :JEE Mains 2023 Hard
- In a binomial distribution \(B ( n , p )\), the sum and product of the mean and variance are \(5\) and \(6\) respectively, then find \(6(n+p-q)\) is equal to :-JEE Mains 2023 Hard
- Let \(S=\left\{\theta \in[-\pi, \pi]-\left\{\pm \frac{\pi}{2}\right\}: \sin \theta \tan \theta+\tan \theta=\sin 2 \theta\right\} \text {. }\) If \(T =\sum_{\theta \in S } \cos 2 \theta\), then \(T + n ( S )\) is equalJEE Mains 2022 Hard
- A possible value of \(^{\prime}x^{\prime}\), for which the ninth term in the expansion of \(\left\{3^{\log _{3} \sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}\right\}^{10}\) in the increasing powers of \(3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}\) is equal to \(180\) , is:JEE Mains 2021 Hard
- Let \(\mathrm{P}(\alpha, \beta, \gamma)\) be the image of the point \(\mathrm{Q}(1,6,4)\) in the line \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\). Then \(2 \alpha+\beta+\gamma\) is equal to ..............JEE Mains 2024 Medium