Conic Sections | Class 11 Mathematics Notes

10.3 Parabola

A parabola is defined as the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. The focus lies in the plane but not on the directrix. The axis of the parabola is the line through the focus perpendicular to the directrix. The vertex is the midpoint between the focus and the directrix. The parabola can be derived geometrically and its equation can be obtained analytically. For example, with the focus at (a, 0) and directrix x = –a, the parabola's equation is y² = 4ax. The latus rectum is a line segment perpendicular to the axis of the parabola through the focus, with endpoints on the parabola. Its length is 4a for the parabola y² = 4ax. The parabola has many real-world applications such as in the design of satellite dishes and car headlights, where the reflective property of parabolas is used to focus signals or light. The chapter also discusses the degenerate case when the focus lies on the directrix, resulting in a straight line.

📊 Diagram: Figure 10: Fig 10.13). (‘Para’ means ‘for’ and ‘bola’ means ‘throwing’, i.e., the shape described when you throw a ball in the air); Table on page 7 (2×2); Figure 11: We will derive the equation for the parabola shown above in Fig 10.15 (a) with; Figure 12: Fig 11.15 (b) as y2 = – 4ax,; Figure 13: To find the Length of the latus rectum of the parabola y2 = 4ax (Fig 10.18).

🧪 Activity: No specific activity, but derivation of parabola equation is a key learning process.

🔗 Connection: Understanding the parabola leads to the study of ellipses, which generalize the concept of distance sums to two foci.

Table on page 7 (2×2)

| 10.4 Parabola Definition 2 A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane. The fixed line is called the directrix of | |

| the parabola and the fixed point F is called the focus (Fig 10.13). (‘Para’ means ‘for’ and ‘bola’ means ‘throwing’, i.e., the shape described when you throw a ball in the air). A Note If the fixed point lies on the fixed line, then the set of points in the plane, which Fig 10. 13 are equidistant from the fixed point and the fixed line is the straight line through the fixed point and perpendicular to the fixed line. We call this straight line as degenerate case of the parabola. A line through the focus and perpendicular to the directrix is called the axis of the parabola. The point of intersection of parabola with the axis is called the vertex of the parabola (Fig10.14). 10.4.1 Standard equations of parabola Fig 10.14 The equation of a parabola is simplest if the vertex is at the origin and the axis of symmetry is along the x-axis or y-axis. The four possible such orientations of parabola are shown below in Fig10.15 (a) to (d). | |

Table on page 8 (1×3)

| Fig 10.15 (a) to (d) We will derive the equation for the parabola shown above in Fig 10.15 (a) with focus at (a, 0) a > 0; and directricx x = – a as below: Let F be the focus and l the directrix. Let FM be perpendicular to the directrix and bisect FM at the point O. Produce MO to X. By the definition of parabola, the mid-point O is on the parabola and is called the vertex of the parabola. Take O as origin, OX the x-axis and OY perpendicular to it as the y-axis. Let the distance from the directrix to the focus be 2a. Then, the coordinates of the focus are (a, 0), and the equation of the directrix is x + a = 0 as in Fig10.16. Let P(x, y) be any point on the parabola such that Fig 10.16 PF = PB, ... (1) where PB is perpendicular to l. The coordinates of B are (– a, y). By the distance formula, we have PF = (x – a)2 + y2 and PB = (x+a)2 Since PF = PB, we have (x – a)2 + y2 = (x+a)2 | | |