JEE Mains · Maths · STD 12 - 11. three dimension geometry
Let the equation of plane passing through the line of intersection of the planes \(x+2 y+a z=2\) and \(x-y+z=3\) be \(5 x-11 y+b z=6 a-1\). For \(c \in Z\), if the distance of this plane from the point \((a,-c, c)\) is \(\frac{2}{\sqrt{a}}\), then \(\frac{a+b}{c}\) is equal to
- A \(-2\)
- B \(2\)
- C \(-4\)
- D \(4\)
Answer & Solution
Correct Answer
(C) \(-4\)
Step-by-step Solution
Detailed explanation
\((x+2 y+a z-2)+\lambda(x-y+z-3)=0\) \(\frac{1+\lambda}{5}=\frac{2-\lambda}{-11}=\frac{ a +\lambda}{ b }=\frac{2+3 \lambda}{6 a -1}\) \(\lambda=-\frac{7}{2}, a =3, b =1\) \(\frac{2}{\sqrt{a}}=\left|\frac{5 a +11 c + bc -6 a +1}{\sqrt{25+121+1}}\right|\) \(c =-1\)…
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 mean and variance of the frequency distribution
are \(9\) and \(15.08\) respectively, then the value of \(\alpha^2+\beta^2-\alpha \beta\) is \(............\).\(x_i\) \(2\) \(4\) \(6\) \(8\) \(10\) \(12\) \(14\) \(16\) \(f_i\) \(4\) \(4\) \(\alpha\) \(15\) \(8\) \(\beta\) \(4\) \(5\) JEE Mains 2023 Hard - If the value of the integral \(\int_{0}^{5} \frac{x+[x]}{e^{x-[x]}} \,d x=\alpha e^{-1}+\beta\) where \(\alpha, \beta \in R, 5 \alpha+6 \beta=0\), and \([\mathrm{x}]\) denotes the greatest integer less than or equal to \(x\); then the value of \((\alpha+\beta)^{2}\) is equal to :JEE Mains 2021 Hard
- Let \(A=I_2-2 \mathrm{MM}^{\mathrm{T}}\), where \(\mathrm{M}\) is real matrix of order \(2 \times 1\) such that the relation \(M^T M=I_1\) holds. If \(\lambda\) is a real number such that the relation \(\mathrm{AX}=\lambda \mathrm{X}\) holds for some non-zero real matrix \(X\) of order \(2 \times 1\), then the sum of squares of all possible values of \(\lambda\) is equal to :JEE Mains 2024 Hard
- The equation \(x^2-4 x+[x]+3=x[x]\), where \([x]\) denotes the greatest integer function, has:JEE Mains 2023 Hard
- The area of the region described by \(A=\{(x,y):x^2 + y^2 \le 1\,and\,y^2 \le 1-x \}\) isJEE Mains 2014 Hard
- A rod of length eight units moves such that its ends \(A\) and \(B\) always lie on the lines \(x-y+2=0\) and \(y+2=0\), respectively. If the locus of the point \(P\), that divides the rod \(A B\) internally in the ratio \(2: 1\) is \(9\left(x^2+\alpha y^2+\beta x y+\gamma x+28 y\right)-76=0\), then \(\alpha-\beta-\gamma\) is equal to :JEE Mains 2025 Hard
More PYQs from JEE Mains
- Let \(e\) be the base of natural logarithm and let \(f: \{1, 2, 3, 4\} \rightarrow \{1, e, e^2, e^3\}\) and \(g: \{1, e, e^2, e^3\} \rightarrow \left\{1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}\right\}\) be two bijective functions such that \(f\) is strictly decreasing and \(g\) is strictly increasing. If \(\phi(x) = \left[f^{-1}\left\{g^{-1}\left(\dfrac{1}{2}\right)\right\}\right]^x\), then the area of the region \(R = \{(x, y): x^2 \leq y \leq \phi(x), 0 \leq x \leq 1\}\) is:JEE Mains 2026 Hard
- Bag \(A\) contains \(2\) white, \(1\) black and \(3\) red balls and bag \(B\) contains \(3\) black, \(2\) red and \(n\) white balls. One bag is chosen at random and \(2\) balls drawn from it at random, are found to be \(1\) red and \(1\) black. If the probability that both balls come from Bag \(A\) is \(\frac{6}{11}\), then \(n\) is equal toJEE Mains 2022 Hard
- The value of \(\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^3\) isJEE Mains 2023 Hard
- If the system of equations \(\mathrm{x}+4 \mathrm{y}-\mathrm{z}=\lambda\), \(7 x+9 y+\mu z=-3,5 x+y+2 z=-1\) has infinitely many solutions, then \((2 \mu .+3 \lambda)\) is equal to :JEE Mains 2024 Medium
- Let \(\alpha|\mathrm{x}|=|\mathrm{y}| \mathrm{e}^{\mathrm{xy}-\beta}, \alpha, \beta \in \mathrm{N}\) be the solution of the differential equation \(x d y-y d x+x y(x d y+y d x)=0\), \(y(1)=2\). Then \(\alpha+\beta\) is equal to ...........JEE Mains 2024 Hard
- Let a function \(f: R \rightarrow R\) be defined as :
\(f(x)=\left\{\begin{array}{ll} \int_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4
\end{array}\right.\)
where \(b \in R\). If \(f\) is continuous at \(x=4\), then which of the following statements is NOT true?JEE Mains 2022 Hard