Chapter 10: Conic Sections

10.1 Introduction

The curves we obtain by slicing a double-napped right circular cone with a plane are called conic sections. Depending on the angle of the intersecting plane relative to the axis of the cone, we get circles, ellipses, parabolas, and hyperbolas.

10.2 Sections of a Cone

Let \( \alpha \) be the angle made by the generator with the axis of the cone, and \( \beta \) be the angle made by the intersecting plane with the axis.

• Circle: When \( \beta = 90^\circ \), the section is a circle.
• Ellipse: When \( \alpha < \beta < 90^\circ \), the section is an ellipse.
• Parabola: When \( \beta = \alpha \), the section is a parabola.
• Hyperbola: When \( 0 \le \beta < \alpha \), the plane intersects both nappes, forming a hyperbola.

10.3 Circle

A circle is the set of all points in a plane that are equidistant from a fixed point in the plane. The fixed point is the center and the fixed distance is the radius.

Equation of a circle with center \( (h, k) \) and radius \( r \): \[ (x - h)^2 + (y - k)^2 = r^2 \]

10.4 Parabola

A parabola is the set of all points in a plane that are equidistant from a fixed line (the directrix) and a fixed point (the focus).

• Vertex: The point of intersection of the parabola with its axis of symmetry.
• Standard Equations:
  1. \( y^2 = 4ax \) (Opens right)
  2. \( y^2 = -4ax \) (Opens left)
  3. \( x^2 = 4ay \) (Opens up)
  4. \( x^2 = -4ay \) (Opens down)
• Latus Rectum: The focal chord perpendicular to the axis of the parabola. Its length is \( 4a \).

10.5 Ellipse

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points (the foci) in the plane is a constant.

• Standard Equation: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
• Vertices: \( (\pm a, 0) \)
• Foci: \( (\pm c, 0) \) where \( c^2 = a^2 - b^2 \)
• Eccentricity: \( e = \frac{c}{a} < 1 \)
• Length of Latus Rectum: \( \frac{2b^2}{a} \)

10.6 Hyperbola

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points (the foci) in the plane is a constant.

• Standard Equation: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• Vertices: \( (\pm a, 0) \)
• Foci: \( (\pm c, 0) \) where \( c^2 = a^2 + b^2 \)
• Eccentricity: \( e = \frac{c}{a} > 1 \)

Competency-Based Questions

Question 1:

Find the equation of a circle passing through the points \( (4,1) \) and \( (6,5) \) and whose center is on the line \( 4x + y = 16 \).

Answer 1:

Let the equation of the circle be \( (x-h)^2 + (y-k)^2 = r^2 \).

Since the center \( (h, k) \) lies on the line \( 4x + y = 16 \), we have:

\[ 4h + k = 16 \Rightarrow k = 16 - 4h \qquad \text{— (1)} \]

The circle passes through \( (4,1) \):

\[ (4-h)^2 + (1-k)^2 = r^2 \qquad \text{— (2)} \]

The circle passes through \( (6,5) \):

\[ (6-h)^2 + (5-k)^2 = r^2 \qquad \text{— (3)} \]

Equating (2) and (3) since both equal \( r^2 \):

\[ (4-h)^2 + (1-k)^2 = (6-h)^2 + (5-k)^2 \]

\[ 16 - 8h + h^2 + 1 - 2k + k^2 = 36 - 12h + h^2 + 25 - 10k + k^2 \]

Cancel \( h^2 \) and \( k^2 \):

\[ 17 - 8h - 2k = 61 - 12h - 10k \]

Bring all terms to one side:

\[ 4h + 8k = 44 \Rightarrow h + 2k = 11 \qquad \text{— (4)} \]

Substitute value of \( k \) from (1) into (4):

\[ h + 2(16 - 4h) = 11 \]

\[ h + 32 - 8h = 11 \]

\[ -7h = -21 \Rightarrow h = 3 \]

Now find \( k \):

\[ k = 16 - 4(3) = 16 - 12 = 4 \]

The center is \( (3, 4) \).

Find radius squared \( r^2 \) using eq (2):

\[ r^2 = (4-3)^2 + (1-4)^2 = 1^2 + (-3)^2 = 1 + 9 = 10 \]

The equation of the circle is \( (x-3)^2 + (y-4)^2 = 10 \), which can be expanded as:

\[ x^2 - 6x + 9 + y^2 - 8y + 16 = 10 \]

\[ x^2 + y^2 - 6x - 8y + 15 = 0 \]

Conclusion: The equation of the circle is \( x^2 + y^2 - 6x - 8y + 15 = 0 \).

Question 2:

Determine the coordinates of the focus, axis, the equation of directrix and latus rectum of the parabola \( y^2 = -8x \).

Answer 2:

The equation involves \( y^2 \), so its axis is the x-axis. The coefficient of \( x \) is negative, meaning the parabola opens towards the left.

Compare \( y^2 = -8x \) with the standard form \( y^2 = -4ax \).

\[ -4a = -8 \Rightarrow a = 2 \]

• Focus: Since it opens to the left, focus \( F(-a, 0) \rightarrow (-2, 0) \).

• Axis: The axis of symmetry is the x-axis, whose equation is \( y = 0 \).

• Directrix: The equation is \( x = a \rightarrow x = 2 \).

• Latus Rectum: Length \( = 4a = 4(2) = 8 \).

Conclusion: Focus: \( (-2, 0) \), Axis: \( y=0 \), Directrix: \( x=2 \), Latus Rectum: 8.