JEE Mains · Maths · STD 12 - 11. three dimension geometry
Let the equation of the plane, that passes through the point \((1,4,-3)\) and contains the line of intersection of the planes \(3 x-2 y+4 z-7=0\) and \(x+5 y-2 z+9=0\), be \(\alpha x+\beta y+\gamma z+3=0\), then \(\alpha+\beta+\gamma\) is equal to :
- A \(-23\)
- B \(-15\)
- C \(23\)
- D \(15\)
Answer & Solution
Correct Answer
(A) \(-23\)
Step-by-step Solution
Detailed explanation
Equation of plane is \(3 x-2 y+4 z-7+\lambda(x+5 y-2 z+9)=0\) \((3+\lambda) x+(5 \lambda-2) y+(4-2 \lambda) z+9 \lambda-7=0\) passing through \((1,4,-3)\) \(\Rightarrow 3+\lambda+20 \lambda-8-12+6 \lambda+9 \lambda-7=0\) \(\Rightarrow \lambda=\frac{2}{3}\)…
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 points on the curve \(y=54 x^5-\) \(135 x^4-70 x^3+180 x^2+210 x\) at which the normal lines are parallel to \(x+90 y+2=0\) is :JEE Mains 2023 Hard
- The sum of the first \(20\) terms of the series \(1 + \frac{3}{2} + \frac{7}{4} + \frac{{15}}{8} + \frac{{31}}{{16}} + ...\) is?JEE Mains 2018 Hard
- If the solution \(y=y(x)\) of the differential equation \(\left(\mathrm{x}^4+2 \mathrm{x}^3+3 \mathrm{x}^2+2 \mathrm{x}+2\right) \mathrm{dy}-\left(2 \mathrm{x}^2+2 \mathrm{x}+3\right) \mathrm{dx}=0\) satisfies \(y(-1)=-\frac{\pi}{4}\), then \(y(0)\) is equal to :JEE Mains 2024 Medium
- Let a complex number be \(w =1-\sqrt{3} i\). Let another complex number \(z\) be such that \(|z w|=1\) and \(\arg ( z )-\arg ( w )=\frac{\pi}{2} .\) Then the area of the triangle with vertices origin, \(z\) and \(w\) is equal to ........ .JEE Mains 2021 Medium
- The number of \( 3\times2 \) matrices A, which can be formed using the elements of the set \( \{-2,-1,0,1,2\} \) such that the sum of all the diagonal elements of \( A^{T}A \) is 5, is ___ .JEE Mains 2026 Hard
- If the vector \(\vec b = 3\hat j + 4\hat k\) is written as the sum of a vector \({\vec {b_1}}\) , parallel to \(\vec a = \hat i + \hat j\) and a vector \({\vec {b_2}}\) , perpendicular to \(\vec a\) , then \({\vec {b_1}} \times {\vec {b_2}}\) is equal toJEE Mains 2017 Hard
More PYQs from JEE Mains
- Let the tangent to the curve \(x^2+2 x-4 y+9=0\) at the point \(P (1,3)\) on it meet the \(y\)-axis at \(A\). Let the line passing through \(P\) and parallel to the line \(x -\) \(3 y=6\) meet the parabola \(y^2=4 x\) at \(B\). If \(B\) lies on the line \(2 x-3 y=8\). then \((A B)^2\) is equal to \(............\).JEE Mains 2023 Hard
- The area of the region \(\left\{(x, y): y^2 \leq 4 x, x<4, \frac{x y(x-1)(x-2)}{(x-3)(x-4)}>0, x \neq 3\right\}\) isJEE Mains 2024 Hard
- If one end of a focal chord of the parabola, \(y^2 = 16x\) is at \((1, 4),\) then the length of this focal chord isJEE Mains 2019 Hard
- If \(\mathrm{U}_{\mathrm{n}}=\left(1+\frac{1}{\mathrm{n}^{2}}\right)\left(1+\frac{2^{2}}{\mathrm{n}^{2}}\right)^{2} \ldots\left(1+\frac{\mathrm{n}^{2}}{\mathrm{n}^{2}}\right)^{\mathrm{n}}\), then \(\lim _{n \rightarrow \infty}\left(U_{n}\right)^{\frac{-4}{n^{2}}}\) is equal to :JEE Mains 2021 Hard
- If the system of linear equations \(x - 2y + kz = 1\) ; \(2x + y + z = 2\) ; \(3x - y - kz = 3\) Has a solution \((x, y, z) \ne 0\), then \((x, y)\) lies on the straight line whose equation isJEE Mains 2019 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