Let’s break down **each part** of the problem for the cone: \[ \{(x, y, z) \in \mathbb{R}^3 \mid 4x^2 + 4y^2 - z^2 = \} \] --- ## 1. **Show it's a Ruled Surface (Parametrization)** A **ruled surface** is a surface that can be swept out by moving a straight line. ### **Rewriting the Equation** \[ 4x^2 + 4y^2 - z^2 = \implies z^2 = 4(x^2 + y^2) \implies z = 2\sqrt{x^2 + y^2} \text{ or } z = -2\sqrt{x^2 + y^2} \] ### **Parametric Representation as a Ruled Surface** Let’s parametrize the cone using a fixed point (vertex at the origin) and a direction. Let’s set \( y = t x \) (so for every \( t \), we get a line in the \( x-y \) plane through the origin): Let \( x = u \), \( y = t u \), \( z = 2\sqrt{x^2 + y^2} = 2|u|\sqrt{1 + t^2} \) But to keep it linear in \( u \), allow \( u \) to be any real number (positive or negative): \[ (x, y, z) = (u, tu, 2u\sqrt{1 + t^2}) \] for all \( u \in \mathbb{R} \), \( t \in \mathbb{R} \). This is a ruled surface: for each value of \( t \), the set of all points as \( u \) varies gives a line passing through the origin (vertex of the cone) and lying on the cone. Alternatively, you can use a trigonometric parametrization: \[ x = r\cos\theta,\, y = r\sin\theta,\, z = 2r \] for \( r \in \mathbb{R} \), \( \theta \in [, 2\pi) \). --- ## 2. **Show the Origin \((,,)\) is the Only Singular Point** A **singular point** on a surface defined by \( F(x, y, z) = \) is where the gradient \( \nabla F = \). Here, \( F(x, y, z) = 4x^2 + 4y^2 - z^2 \): \[ \nabla F = (8x,\, 8y,\, -2z) \] Set each partial derivative to : - \( 8x = \implies x = \) - \( 8y = \implies y = \) - \( -2z = \implies z = \) So, **the only singular point is \((, , )\)**. --- ## 3. **Find the "Line" of Striction** The **line of striction** of a ruled surface is the locus of the closest points between adjacent rulings (lines). For a cone with vertex at the origin, **the line of striction is the vertex itself**, i.e., the origin. But if you want the more general answer: - For a cone, all rulings pass through the vertex (origin). - Thus, the "line" of striction degenerates to the **point (,,)**. --- ## **Summary** - **Ruled surface parametrization:** \((x, y, z) = (u, tu, 2u\sqrt{1 + t^2})\) - **Only singular point:** \((,,)\) - **Line of striction:** The origin \((,,)\) Let me know if you want a more geometric or visual explanation!