JEE Mains · Maths · STD 12 - 11. three dimension geometry
The equation of the planes parallel to the plane \(x\) \(-2 y +2 z -3=0\) which are at unit distance from the point \((1,2,3)\) is \(a x+b y+c z+d=0\). If \((b-d)=K(c-a),\) then the positive value of \(K\) is
- A \(4\)
- B \(6\)
- C \(2\)
- D \(1\)
Answer & Solution
Correct Answer
(A) \(4\)
Step-by-step Solution
Detailed explanation
Let plane is \(x-2 y+2 z+\lambda=0\) distance from \((1,2,3)=1\) \(\Rightarrow \frac{|\lambda+3|}{5}=1 \Rightarrow \lambda=0,-6\) \(\Rightarrow a =1, b =-2, c =2, d =-6\) or \(0\) \(b - d =4\) or \(-2, c - a =1\) \(\Rightarrow k =4\) or \(-2\)
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
- Three points \(\mathrm{O}(0,0), \mathrm{P}\left(\mathrm{a}, \mathrm{a}^2\right), \mathrm{Q}\left(-\mathrm{b}, \mathrm{b}^2\right), \mathrm{a}>0, \mathrm{~b}>0\), are on the parabola \(y=x^2\). Let \(S_1\) be the area of the region bounded by the line \(P Q\) and the parabola, and \(S_2\) be the area of the triangle \(O P Q\). If the minimum value of \(\frac{\mathrm{S}_1}{\mathrm{~S}_2}\) is \(\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\), then \(\mathrm{m}+\mathrm{n}\) is equal to :JEE Mains 2024 Hard
- If the surface area of a cube is increasing at a rate of \(3.6 cm ^{2} / sec ,\) remaining its shape; then the rate of change of its volume (in \(cm ^{3} / sec\) ), when the length of a side of the cube is \(10 cm ,\) isJEE Mains 2020 Medium
- Let the matrix \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]\) satisfy \(A^n=A^{n-2}+A^2-I\) for \(\mathrm{n} \geq 3\). Then the sum of all the elements of \(\mathrm{A}^{50}\) is :-JEE Mains 2025 Medium
- Let \(\overrightarrow{ a }=2 \hat{ i }-3 \hat{ j }+4 \hat{ k }\) and \(\overrightarrow{ b }=7 \hat{ i }+\hat{ j }-6 \hat{ k }\) If \(\overrightarrow{ r } \times \overrightarrow{ a }=\overrightarrow{ r } \times \overrightarrow{ b }, \overrightarrow{ r } \cdot(\hat{ i }+2 \hat{ j }+\hat{ k })=-3,\) then \(\overrightarrow{ r } \cdot(2 \hat{ i }-3 \hat{ j }+\hat{ k })\) is equal toJEE Mains 2021 Hard
- Let \( \vec{a}=\hat{i}-2\hat{j}+3\hat{k},\vec{b}=2\hat{i}+\hat{j}-\hat{k},\vec{c}=\lambda\hat{i}+\hat{j}+\hat{k} \) and \( \vec{v}=\vec{a}\times\vec{b} \). If \( \vec{v} \cdot \vec{c}=11 \) and the length of the projection of \( \vec{b} \) on \( \vec{c} \) is \( p \), then \( 9p^{2} \) is equal to:JEE Mains 2026 Medium
- Let \(f(\mathrm{x})=\mathrm{x} \cos ^{-1}(-\sin |\mathrm{x}|), \quad \mathrm{x} \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right],\) then which of the following is true?JEE Mains 2020 Hard
More PYQs from JEE Mains
- If the function \(f\) defined as \(f(x)\, = \frac{1}{x} - \frac{{k - 1}}{{{e^{2x}} - 1}}\) ,\(x\, \ne \,0,\) is continuous at \(x = 0.\) then the ordered pair \((k,f(0))\) is equal to?JEE Mains 2018 Hard
- Let \(g(x)=f(x)+f(1-x)\) and \(f^{\prime \prime}(x) > 0, x \in(0,1)\). If \(g\) is decreasing in the interval \((0, \alpha)\) and increasing in the interval \((\alpha, 1)\), then \(\tan ^1(2 \alpha)+\tan ^{-1}\left(\frac{1}{\alpha}\right)+\tan ^{-1}\left(\frac{\alpha+1}{\alpha}\right)\) is equal to :JEE Mains 2023 Hard
- If the point of intersection of the lines \(\dfrac{x+1}{3} = \dfrac{y+a}{5} = \dfrac{z+b+1}{7}\) and \(\dfrac{x-2}{1} = \dfrac{y-b}{4} = \dfrac{z-2a}{7}\) lies on \(xy\)-plane, then the value of \(a + b\) is :JEE Mains 2026 Medium
- The sum of the distinct real values of \(\mu \), for which the vectors, \(\mu \hat i + \hat j + \hat k,\,\hat i + \mu \hat j + \hat k,\,\hat i + \hat j + \mu \hat k\) are co-planar, isJEE Mains 2019 Medium
- Consider a set of \(3 n\) numbers having variance \(4.\) In this set, the mean of first \(2 n\) numbers is \(6\) and the mean of the remaining \(n\) numbers is \(3.\) A new set is constructed by adding \(1\) into each of first \(2 n\) numbers, and subtracting \(1\) from each of the remaining \(n\) numbers. If the variance of the new set is \(k\), then \(9 k\) is equal to .... .JEE Mains 2021 Hard
- Let \(y = y ( x )\) be the solution of the differential equation \(x d y-y d x=\sqrt{\left(x^{2}-y^{2}\right)} d x, x \geq 1\), with \(y (1)=0 .\) If the area bounded by the line \(x =1, x = e ^{\pi}, y =0\) and \(y = y ( x )\) is \(\alpha e ^{2 \pi}+\beta\) then the value of \(10(\alpha+\beta)\) is equal to ....... .JEE Mains 2021 Medium