Discussion
The locus of points of intersection of two surfaces represented by the partial differential equations \(x\,dx + y\,dy + z\,dz = 0\) and \(yp - xq = 0\) is a :
Answer: B
The intersections are circles.
The Pfaffian equation \(x\,dx + y\,dy + z\,dz = 0\) integrates to \(x^{2} + y^{2} + z^{2} = c_{1}\): spheres centred at the origin.
The linear equation \(yp - xq = 0\) has Lagrange subsidiary equations \(dx/y = dy/(-x) = dz/0\), giving \(z = \text{const}\) and \(x^{2} + y^{2} = \text{const}\). Its integral surfaces are \(z = \phi(x^{2} + y^{2})\), surfaces of revolution about the \(z\)-axis.
A sphere centred on the axis meets a surface of revolution about that axis in curves that are themselves symmetric about the axis and lie on the sphere: horizontal circles \(z = \text{const}\), \(x^{2} + y^{2} = \text{const}\). The characteristics of \(yp - xq = 0\) are exactly such circles. So the locus is not a parabola, straight line or helix (a helix is not symmetric under rotation about its axis).
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development