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Conic Sections - Degenerated conic sections

Grade 11CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A conic section is formed by the intersection of a plane with a double-napped right circular cone. When the plane passes through the vertex of the cone, the resulting section is called a degenerate conic section.

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Let α\alpha be the angle between the axis of the cone and the generator, and β\beta be the angle made by the intersecting plane with the axis of the cone.

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Case 1: When β>α\beta > \alpha, the plane contains only the vertex of the cone. The degenerate conic is a single point.

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Case 2: When β=α\beta = \alpha, the plane is tangent to the cone along a generator. The degenerate conic is a single straight line.

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Case 3: When 0≤β<α0 \le \beta < \alpha, the plane cuts both nappes of the cone and passes through the vertex. The degenerate conic is a pair of intersecting lines.

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In the general second-degree equation Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, the conic is degenerate if the determinant of the matrix associated with the quadratic form and linear parts is zero.

📐Formulae

x2+y2=0(Represents a single point at the origin)x^2 + y^2 = 0 \quad \text{(Represents a single point at the origin)}

x2−k2y2=0  ⟹  (x−ky)(x+ky)=0(Represents two intersecting lines)x^2 - k^2 y^2 = 0 \implies (x - ky)(x + ky) = 0 \quad \text{(Represents two intersecting lines)}

x2=0(Represents a single line, specifically the y-axis coincident with itself)x^2 = 0 \quad \text{(Represents a single line, specifically the y-axis coincident with itself)}

Condition for intersecting lines: x2a2−y2b2=0\text{Condition for intersecting lines: } \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0

💡Examples

Problem 1:

Identify the type of degenerate conic represented by the equation 9x2−16y2=09x^2 - 16y^2 = 0.

Solution:

The given equation is 9x2−16y2=09x^2 - 16y^2 = 0. This can be written as (3x)2−(4y)2=0(3x)^2 - (4y)^2 = 0. Using the identity a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b), we get (3x−4y)(3x+4y)=0(3x - 4y)(3x + 4y) = 0. This represents two distinct lines: 3x−4y=03x - 4y = 0 and 3x+4y=03x + 4y = 0.

Explanation:

Since the equation resolves into two linear factors that intersect at the origin (0,0)(0,0), it represents a pair of intersecting lines.

Problem 2:

What geometric figure is represented by the equation x2+4y2=0x^2 + 4y^2 = 0 in the real number system?

Solution:

In the equation x2+4y2=0x^2 + 4y^2 = 0, both x2x^2 and 4y24y^2 are non-negative for all real values of xx and yy. The only way their sum can be zero is if x2=0x^2 = 0 and 4y2=04y^2 = 0. This implies x=0x = 0 and y=0y = 0.

Explanation:

Because the only real solution is the coordinate (0,0)(0,0), the equation represents a degenerate conic which is a single point.

Problem 3:

Describe the intersection of a plane and a cone when the plane passes through the vertex and the angle β\beta is equal to the semi-vertical angle α\alpha.

Solution:

When the plane passes through the vertex and β=α\beta = \alpha, the plane is exactly parallel to the generator of the cone and touches the cone along that generator line.

Explanation:

In this specific geometric orientation, the intersection is not a curve (like a parabola) but a single straight line passing through the vertex.