Q1.. Check whether the following statements are true or false. Justify your answer with a short explanation or a counter example.
- (i)) Any line through the origin cuts the sphere x² + y² + z² = 4 at exactly two points. (50 words)
- (ii)) The plane making intercept at the z-axis and parallel to the xy-plane intersects the cone x² + y² = z²tan²θ in a circle. (50 words)
- (iii)) There exists no line with 1/2, 1/2, 1/2 as direction cosines. (50 words)
- (iv)) The tangent planes at the extremities of any axis of an ellipsoid are perpendicular. (50 words)
- (v)) A section of an elliptic paraboloid by a plane is always an ellipse. (50 words)
- (vi)) The curve xy² + yx² = 0 is symmetric about the origin. (50 words)
- (vii)) There exists a unique line which is perpendicular to the lines x = y = z/2 and x = y = -z. (50 words)
- (viii)) The plane 3x + 4y + 2z = 1 touches the conicoid 3x² + 2y² = z² = 1. (50 words)
- (ix)) The xy- plane intersects the sphere x² + y² + z² + 2x − z = 2 in a great circle. (50 words)
- (x)) Non degenerate conics are non-central. (50 words)
Key Points:
- Direction Cosines: Sum of squares (l²+m²+n²) must equal 1.
- Sphere-Line Intersection: A line through a sphere's center always cuts it at two points.
- Conicoid Tangency: Plane Ax+By+Cz=D touches ax²+by²+cz²=1 if A²/a+B²/b+C²/c = D².
- Great Circle: Intersection when a plane passes through the sphere's center.
Answer: This response comprehensively analyzes ten statements related to Analytical Geometry from the MTE-05 course. Each statement is evaluated as true or false, with a concise justification provided, often including relevant definitions, theorems, or counter-examples. The explanations adhere to the specified word limits for each sub-question, ensuring clarity and precision in accordance with the principles of analytical geometry.