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Module 1: Advanced Euclidean Geometry

Topics Covered:

1. Axioms and Theorems in Euclidean Geometry
2. Properties of Parallel Lines with Advanced Proofs
3. Advanced Properties of Polygons:
   • Sum of Interior and Exterior Angles
   • Regular and Irregular Polygons

Practice Questions:

1. Prove: The sum of the exterior angles of any polygon is 360∘360^\circ360∘.
2. If a regular polygon has 12 sides, find the measure of one interior angle.
3. Prove that two lines parallel to the same line are parallel to each other.

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Module 2: Advanced Circle Theorems

Topics Covered:

1. Tangents to a Circle:
   • Properties of Tangents
   • Tangents from an External Point
2. Angles in a Circle:
   • Angles in the Same Segment
   • Angle at the Center and at the Circumference
3. Advanced Cyclic Quadrilateral Properties

Practice Questions:

1. Prove that the tangents drawn from an external point to a circle are equal in length.
2. In a cyclic quadrilateral, prove that the opposite angles are supplementary.
3. A chord is bisected by a radius at a right angle. Prove that the radius passes through the center.

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Module 3: Conic Sections

Topics Covered:

1. Parabola:
   • Standard Equation and Focus-Directrix Property
2. Ellipse:
   • Standard Equations and Eccentricity
3. Hyperbola:
   • Properties and Equations

Practice Questions:

1. Derive the equation of a parabola with the focus at (0,2)(0, 2)(0,2) and the directrix y=−2y = -2y=−2.
2. Find the eccentricity of an ellipse with the equation x225+y216=1\frac{x^2}{25} + \frac{y^2}{16} = 125×2​+16y2​=1.
3. Sketch the graph of x29−y24=1\frac{x^2}{9} – \frac{y^2}{4} = 19×2​−4y2​=1 and identify its asymptotes.

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Module 4: Advanced Coordinate Geometry

Topics Covered:

1. Equation of a Circle:
   • General Form
   • Tangents and Normals
2. Locus and Its Equations
3. Advanced Applications:
   • Area of Triangles and Quadrilaterals
   • Collinearity of Points Using Determinants

Practice Questions:

1. Find the equation of a circle with center (2,−3)(2, -3)(2,−3) and radius 555.
2. Determine the locus of a point that is equidistant from (1,2)(1, 2)(1,2) and (4,6)(4, 6)(4,6).
3. Prove that the points (2,3),(4,5),(−1,−2)(2, 3), (4, 5), (-1, -2)(2,3),(4,5),(−1,−2) are collinear using determinants.

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Module 5: 3D Geometry

Topics Covered:

1. Introduction to 3D Coordinate Geometry:
   • Distance Formula in 3D
   • Section Formula in 3D
2. Equation of a Plane:
   • General and Intercept Forms
3. Angle Between Two Planes and Lines

Practice Questions:

1. Find the distance between two points (1,2,3)(1, 2, 3)(1,2,3) and (4,6,8)(4, 6, 8)(4,6,8) in 3D space.
2. Write the equation of a plane passing through (1,2,3)(1, 2, 3)(1,2,3) and parallel to the XY-plane.
3. Find the angle between two planes 2x+3y+z=52x + 3y + z = 52x+3y+z=5 and x−y+4z=6x – y + 4z = 6x−y+4z=6.

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Module 6: Transformations in Geometry

Topics Covered:

1. Rotations in 2D and 3D
2. Advanced Reflections
3. Homothety and Scaling in 2D and 3D

Practice Questions:

1. Rotate the point (2,−3)(2, -3)(2,−3) about the origin by 90∘90^\circ90∘ counterclockwise.
2. Reflect the point (4,5,−2)(4, 5, -2)(4,5,−2) about the XY-plane.
3. Scale a triangle with vertices (0,0),(2,0),(1,1)(0, 0), (2, 0), (1, 1)(0,0),(2,0),(1,1) by a factor of 2.

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Module 7: Vector Geometry

Topics Covered:

1. Vectors in Geometry:
   • Vector Addition, Subtraction, and Scalar Multiplication
2. Dot Product and Cross Product Applications
3. Geometry of Lines and Planes Using Vectors

Practice Questions:

1. Find the angle between two vectors a=2i+3j+k\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} + \mathbf{k}a=2i+3j+k and b=i−j+4k\mathbf{b} = \mathbf{i} – \mathbf{j} + 4\mathbf{k}b=i−j+4k.
2. Prove that the vectors a=3i−2j\mathbf{a} = 3\mathbf{i} – 2\mathbf{j}a=3i−2j and b=6i−4j\mathbf{b} = 6\mathbf{i} – 4\mathbf{j}b=6i−4j are collinear.
3. Find the equation of a plane passing through the point (1,2,3)(1, 2, 3)(1,2,3) and perpendicular to the vector 2i−j+k2\mathbf{i} – \mathbf{j} + \mathbf{k}2i−j+k.

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Module 8: Advanced Applications of Geometry

Topics Covered:

1. Geometry in Real-Life Problems:
   • Navigation and Distance
   • Construction and Architecture
2. Optimization Problems:
   • Shortest Path Between Two Points
3. Geometry in Physics and Engineering

Practice Questions:

1. A lighthouse is located at (0,0)(0, 0)(0,0) and a ship at (5,12)(5, 12)(5,12). Find the shortest path the ship must take to reach the lighthouse.
2. A cable is stretched from the top of a tower (0,h)(0, h)(0,h) to a point on the ground (a,0)(a, 0)(a,0). Find the length of the cable.
3. Design a rectangular garden with a given area such that the perimeter is minimized. Use geometry principles to justify your design.

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Module 9: Non-Euclidean Geometry

Topics Covered:

1. Introduction to Non-Euclidean Geometries:
   • Hyperbolic and Spherical Geometry
2. Comparison with Euclidean Geometry
3. Applications of Non-Euclidean Geometry

Practice Questions:

1. Explain why the sum of angles in a triangle is less than 180∘180^\circ180∘ in hyperbolic geometry.
2. Calculate the spherical distance between two points on a sphere with a given radius.
3. Compare the parallel postulate in Euclidean and hyperbolic geometries.

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Module 10: Final Review and Advanced Mock Test

Topics Covered:

1. Mixed Questions on All Topics
2. Geometry Project:
   • Analyze the Structural Design of Famous Buildings
   • Create a Model Using Advanced Geometric Principles
3. Mock Test for Competitive Exams

Mock Test Sample Questions:

1. Prove that the diagonals of a rhombus bisect each other at right angles.
2. Derive the standard equation of a hyperbola with foci at (±c,0)(\pm c, 0)(±c,0) and vertices at (±a,0)(\pm a, 0)(±a,0).
3. A line passes through the point (1,2,3)(1, 2, 3)(1,2,3) and is parallel to 2i+3j−k2\mathbf{i} + 3\mathbf{j} – \mathbf{k}2i+3j−k. Write its vector equation.

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Additional Materials

• Graphing Tools for Visualizing Complex Geometry
• Advanced Problem Sets for Competitive Exams
• **Real-Life Case Studies

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