Let A1, A2, A3, A4,........, A8 be the vertices of the regular octagons that lie on the circle of radius 2. Let p be a point on the circle and let PAi denote the distance between the point P and Ai for i = 1,2,3,....,8. If P varies over the circle, then the maximum value of the product is PA1.Pa2..........PA8, is
Correct Answer: 512
Solution and Explanation
The correct answer is: 29 = 512
The vertices A1, A2, A3, ..., A8 of a regular octagon are positioned on a circle with a radius of 2. Consider Z as the result of evaluating (2) × (1) × (1/8), resulting in Z8 being equal to 28 × 1.
Thus, the expression Z8 – 28 equals zero.MAX ( P1,P2,P8)=29 = 512
Top Questions on introduction to three dimensional geometry
Let Q be the cube with the set of vertices {(x1, x2, x3) ∈ R3: x1, x2, x3 ∈ {0,1}}. Let F be the set of all twelve lines containing the diagonals of the six faces of cube Q. Let S be the set of all four lines containing the main diagonals of the cube Q; for instance, the line passing through the vertices (0,0,0) and (1,1,1) is in S. For lines l1 and l2, let d(l1,l2) denote the shortest distance between them. Then the maximum value of d(l1,l2) as l1 varies over f and l2 varies over S, is
- JEE Advanced - 2023
- Mathematics
- introduction to three dimensional geometry
Consider an obtuse-angled triangle ABC in which the difference between the largest and the smallest angle is \(\frac{\pi}{2}\) and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.Then the inradius of the triangle ABC is
- JEE Advanced - 2023
- Mathematics
- introduction to three dimensional geometry
Consider an obtuse-angled triangle ABC in which the difference between the largest and the smallest angle is \(\frac{\pi}{2}\) and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.Let a be the area of the triangle ABC. Then the value of (64a)2 is
- JEE Advanced - 2023
- Mathematics
- introduction to three dimensional geometry
- The equation of conic is \(19x^2 + 15y^2 = 285\). A concentric circle with radius 4 units is given then angle of common tangent made by minor axis of ellipse is
- JEE Main - 2023
- Mathematics
- introduction to three dimensional geometry
- Consider an obtuse-angled triangle ABC in which the difference between the largest and the smallest angle is \(\frac{\pi}{2}\) and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.Let a be the area of the triangle ABC. Then the value of (64a)2 is
- JEE Advanced - 2023
- Mathematics
- introduction to three dimensional geometry
Questions Asked in JEE Advanced exam
- Consider the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\). Let S (p, q) be a point in the first quadrant such that \(\frac{p^2}{9}+\frac{q^2}{4}>1\). Two tangents are drawn from S to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point T in the fourth quadrant. Let R be the vertex of the ellipse with positive x-coordinate and Obe the center of the ellipse. If the area of the triangle △ORT is \(\frac{3}{2}\) , then which of the following options is correct ?
- JEE Advanced - 2024
- Conic sections
- Among V(CO)6, Cr(CO)5, Cu(CO)3, Mn(CO)5, Fe(CO)5, [Co(CO)3]3-, [Cr(CO)4]4-, and Ir(CO)3, the total number of species isoelectronic with Ni(CO)4 is _________.
[Given, atomic number: V = 23, Cr=24, Mn 25, Fe 26, Co 27, Ni 28, Cu = 29, Ir = 77]- JEE Advanced - 2024
- Isomerism in Coordination Compounds
- Let \(\overrightarrow{OP}=\frac{\alpha-1}{\alpha}\hat{i}+\hat{j}+\hat{k},\overrightarrow{OQ}=\hat{i}+\frac{\beta-1}{\beta}\hat{j}+\hat{k}\) and \(\overrightarrow{OR}=\hat{i}+\hat{j}+\frac{1}{2}\hat{k}\) be three vector where α, β ∈ R - {0} and 0 denotes the origin. If \((\overrightarrow{OP}\times\overrightarrow{OQ}).\overrightarrow{OR}=0\) and the point (α, β, 2) lies on the plane 3x + 3y - z + l = 0, then the value of l is _______.
- JEE Advanced - 2024
- Vector Algebra
- A closed vessel contains 10 g of an ideal gas X at 300 K, which exerts 2 atm pressure. At the same temperature, 80 g of another ideal gas Y is added to it and the pressure becomes 6 atm. The ratio of root mean square velocities of X and Y at 300 K is
- JEE Advanced - 2024
- States of matter
- Let the function \(f:[1,\infin)→\R\) be defined by
\(f(t) = \begin{cases} (-1)^{n+1}2, & \text{if } t=2n-1,n\in\N, \\ \frac{(2n+1-t)}{2}f(2n-1)+\frac{(t-(2n-1))}{2}f(2n+1) & \text{if } 2n-1<t<2n+1,n\in\N. \end{cases}\)
Define \(g(x)=\int\limits_{1}^{x}f(t)dt,x\in(1,\infin).\) Let α denote the number of solutions of the equation g(x) = 0 in the interval (1, 8] and \(β=\lim\limits_{x→1+}\frac{g(x)}{x-1}\). Then the value of α + β is equal to _____.- JEE Advanced - 2024
- Integral Calculus
Concepts Used:
Introduction to Three Dimensional Geometry
In mathematics, Geometry is one of the most important topics. The concepts of Geometry are defined with respect to the planes. So, Geometry is divided into three categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line
Let's consider line ‘L’ is passing through the three-dimensional plane. Now, x,y, and z are the axes of the plane, and α,β, and γ are the three angles the line making with these axes. These are called the plane's direction angles. So, correspondingly, we can very well say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
Read More: Introduction to Three-Dimensional Geometry