Conic Sections | Class 11 Mathematics Notes

10.4 Ellipse

An ellipse is defined as the set of all points in a plane such that the sum of the distances from two fixed points called foci is constant. The midpoint of the line segment joining the foci is called the centre of the ellipse. The line segment through the foci is called the major axis, and the line segment through the centre perpendicular to the major axis is called the minor axis. The endpoints of the major axis are called the vertices. The constant sum of distances is always greater than the distance between the two foci. The standard form of the ellipse equation with centre at origin and major axis along the x-axis is (x²/a²) + (y²/b²) = 1, where 2a is the length of the major axis and 2b is the length of the minor axis. The foci lie at (±c, 0) where c² = a² – b². The eccentricity e = c/a measures the deviation of the ellipse from being circular. The latus rectum is a line segment perpendicular to the major axis through a focus, with length 2b²/a. Ellipses appear in planetary orbits and optics.

📊 Diagram: Figure 15: axis. The end points of the major axis are called the vertices of the ellipse(Fig 10.21); Table on page 12 (1×3); Figure 16: (Fig 10.23).; Figure 17: (– c, 0) and F2 be (c, 0) (Fig 10.25); Table on page 14 (2×2); Figure 20: lie on the ellipse (Fig 10.26); Table on page 17 (1×2); Figure 21; Figure 22; Figure 23

🧪 Activity: No specific activity, but examples and derivations illustrate ellipse properties.

🔗 Connection: The ellipse section prepares for the study of hyperbolas, which involve the difference of distances from two foci.

Table on page 12 (1×3)

| 10. 5 Ellipse Definition 4 An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant. The two fixed points are called the foci (plural of ‘focus’) of the ellipse (Fig10.20). A Note The constant which is the sum of the distances of a point on the ellipse from the two fixed points is always greater than the Fig 10.20 distance between the two fixed points. The mid point of the line segment joining the foci is called the centre of the ellipse. The line segment through the foci of the ellipse is called the major axis and the line segment through the centre and perpendicular to the major axis is called the minor axis. The end points of the major axis are called the vertices of the ellipse(Fig 10.21). Fig 10.21 Fig 10.22 | 10. | |

Table on page 14 (2×2)

| (a) Fig 10.24 Let F and F be the foci and O be the mid-point of the line segment F F . Let O 1 2 1 2 be the origin and the line from O through F be the positive 2 x-axis and that through F as the negative x-axis. 1 Let, the line through O perpendicular to the x-axis be the y-axis. Let the coordinates of F be 1 (– c, 0) and F be (c, 0) (Fig 10.25). 2 Let P(x, y) be any point on the ellipse such that the sum of the distances from P to the two foci be 2a so given PF + PF = 2a. ... (1) 1 2 Using the distance formula, we have x2 y2 (x + c)2 + y2 + (x − c)2 + y2 = 2a + =1 a2 b2 Fig 10.25 i.e., (x + c)2 + y2 = 2a – (x−c)2 + y2 Squaring both sides, we get | |

Table on page 17 (1×2)

| Definition 6 Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse (Fig 10.26). | |

| To find the length of the latus rectum x2 y2 of the ellipse + =1 a2 b2 Let the length of AF be l. Fig 10. 26 2 Then the coordinates of A are (c, l ),i.e., (ae, l ) x2 y2 Since A lies on the ellipse + = 1, we have a2 b2 (ae)2 l2 + =1 a2 b2 ⇒ l2 = b2 (1 – e2) c2 a 2 – b2 b2 But e2 = = =1– a2 a2 a2 b4 b2 Therefore l2 = , i.e., l = a2 a Since the ellipse is symmetric with respect to y-axis (of course, it is symmetric w.r.t. 2b2 both the coordinate axes), AF = F B and so length of the latus rectum is . 2 2 a Example 9 Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the latus rectum of the ellipse x2 y2 + =1 25 9 | |