JEE Mains · Maths · STD 12 - 11. three dimension geometry
Let the equation of the plane \(P\) containing the line \(x+10=\frac{8-y}{2}=z\) be \(a x+b y+3 z=2(a+b)\) and the distance of the plane \(P\) from the point \((1,27,7)\) be c. Then \(a^2+b^2+c^2\) is equal to \(.............\).
- A \(354\)
- B \(353\)
- C \(355\)
- D \(35.5\)
Answer & Solution
Correct Answer
(C) \(355\)
Step-by-step Solution
Detailed explanation
The line \(\frac{x+10}{1}=\frac{y-8}{-2}=\frac{z}{1}\) have a point \((-10,8,0)\) with d. r. \((1,-2,1)\) \(\because\) the plane \(a x+b y+3 z=2(a+b)\) \(\Rightarrow b =2 a\) and dot product of d.r.'s is zero \(\therefore a-2 b+3=0\) \(\therefore a =1\) and \(b =2\) 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
- If the four distinct points \((4,6),(-1,5),(0,0)\) and \((\mathrm{k}, 3 \mathrm{k})\) lie on a circle of radius r , then \(10 \mathrm{k}+\mathrm{r}^2\) is equal toJEE Mains 2025 Medium
- The distance, of the point \((7,-2,11)\) from the line \(\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3}\) along the line \(\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6}\), is :JEE Mains 2024 Hard
- The area of the region bounded by the curve \(y=\max \{|x|, x|x-2|\}\), then \(x\)-axis and the lines \(x=-2\) and \(x=4\) is equal to _______ .JEE Mains 2025 Easy
- Let the mean and standard deviation of marks of class \(A\) of \(100\) students be respectively \(40\) and \(\alpha( > 0)\), and the mean and standard deviation of marks of class \(B\) of \(n\) students be respectively \(55\) and \(30-\alpha\). If the mean and variance of the marks of the combined class of \(100+ n\) students are respectively \(50\) and \(350\),then the sum of variances of classes \(A\) and \(B\) isJEE Mains 2023 Hard
- If the orthocentre of the triangle formed by the lines \(2 x+3 y-1=0, x+2 y-1=0\) and \(a x+b y-1=0\), is the centroid of another triangle, whose circumcentre and orthocentre respectively are \((3,4)\) and \((-6,-8)\), then the value of \(|a-b|\) is ........... .JEE Mains 2024 Medium
- If the arithmetic mean and geometric mean of the \(p ^{\text {th }}\) and \(q ^{\text {th }}\) terms of the sequence \(-16,8,-4,2, \ldots\) satisfy the equation \(4 x^{2}-9 x+5=0,\) then \(p+q\) is equal to ..... .JEE Mains 2021 Hard
More PYQs from JEE Mains
- Let \(f(x)=3^{\left(x^{2}-2\right)^{3}+4}, x \in R\). Then which of the following statements are true ? \(P: x=0\) is a point of local minima of \(f\) \(Q: x=\sqrt{2}\) is a point of inflection of \(f\) \(R: f^{\prime}\) is increasing for \(x>\sqrt{2}\)JEE Mains 2022 Hard
- The vector \(\left( {\hat i \times \vec a.\vec b} \right)\hat i + \left( {\hat j \times \vec a.\vec b} \right)\hat j + \left( {\hat k \times \vec a.\vec b} \right)\hat k\) is equal toJEE Mains 2013 Hard
- Let \(A\) be a matrix of order \(3 \times 3\) and det \((A)=2\). Then \(\operatorname{det}\left(\operatorname{det}( A )\right.\) adj \(\left(5 \operatorname{adj}\left( A ^{3}\right)\right)\) ) is equal to.....JEE Mains 2022 Hard
- \(\lim _{x \rightarrow 0^{+}} \frac{\tan \left(5(x)^{\frac{1}{3}}\right) \log _e\left(1+3 x^2\right)}{\left(\tan ^{-1} 3 \sqrt{x}\right)^2\left(e^{5(x)^{\frac{4}{3}}}-1\right)}\) is equal toJEE Mains 2025 Medium
- Let \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) be a thrice differentiable function such that \(f(0)=0, f(1)=1, f(2)=-1, f(3)=2\) and \(f(4)=-2\). Then, the minimum number of zeros of \(\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x)\) is ...........JEE Mains 2024 Hard
- The set of all values of \(k\) for which \(\left(\tan ^{-1} x \right)^{3}+\left(\cot ^{-1} x \right)^{3}= k \pi^{3}, x \in R\), is the intervalJEE Mains 2022 Hard