Chapter 10: Conic Sections

Ans: The given equation is x²/36 + y²/16 = 1.

Here, the denominator of x²/36 is greater than the denominator of y²/16. Therefore, the major axis is along the x-axis, and the minor axis is along the y-axis.

Comparing with the standard equation x²/a² + y²/b² = 1, we get a² = 36 and b² = 16, so a = 6 and b = 4.

Now, c = √(a² − b²) = √(36 − 16) = √20 = 2√5

• Coordinates of the foci: (2√5, 0) and (−2√5, 0)

• Coordinates of the vertices: (6, 0) and (−6, 0)

• Length of major axis = 2a = 12

• Length of minor axis = 2b = 8

• Eccentricity, e = c/a = 2√5/6 = √5/3

• Length of latus rectum = 2b²/a = (2 × 16)/6 = 16/3

Ans: The given equation is x²/4 + y²/25 = 1.

Here, the denominator of y²/25 is greater than the denominator of x²/4. Therefore, the major axis is along the y-axis, and the minor axis is along the x-axis.

Comparing with x²/b² + y²/a² = 1, we get a² = 25 and b² = 4, so a = 5 and b = 2.

Now, c = √(a² − b²) = √(25 − 4) = √21

• Coordinates of the foci: (0, √21) and (0, −√21)

• Coordinates of the vertices: (0, 5) and (0, −5)

• Length of major axis = 2a = 10

• Length of minor axis = 2b = 4

• Eccentricity, e = c/a = √21/5

• Length of latus rectum = 2b²/a = (2 × 4)/5 = 8/5

Ans: The denominator of x²/16 is greater than that of y²/9, so the major axis is along the x-axis.

Comparing with x²/a² + y²/b² = 1, we get a = 4 and b = 3.

Now, c = √(a² − b²) = √(16 − 9) = √7

• Coordinates of the foci: (√7, 0) and (−√7, 0)

• Coordinates of the vertices: (4, 0) and (−4, 0)

• Length of major axis = 2a = 8

• Length of minor axis = 2b = 6

• Eccentricity, e = c/a = √7/4

• Length of latus rectum = 2b²/a = (2 × 9)/4 = 9/2

Ans: The denominator of y²/100 is greater than that of x²/25, so the major axis is along the y-axis.

Comparing with x²/b² + y²/a² = 1, we get a = 10 and b = 5.

Now, c = √(a² − b²) = √(100 − 25) = √75 = 5√3

• Coordinates of the foci: (0, 5√3) and (0, −5√3)

• Coordinates of the vertices: (0, 10) and (0, −10)

• Length of major axis = 2a = 20

• Length of minor axis = 2b = 10

• Eccentricity, e = c/a = 5√3/10 = √3/2

• Length of latus rectum = 2b²/a = (2 × 25)/10 = 5

Ans: The denominator of x²/49 is greater than that of y²/36, so the major axis is along the x-axis.

Comparing with x²/a² + y²/b² = 1, we get a = 7 and b = 6.

Now, c = √(a² − b²) = √(49 − 36) = √13

• Coordinates of the foci: (√13, 0) and (−√13, 0)

• Coordinates of the vertices: (7, 0) and (−7, 0)

• Length of major axis = 2a = 14

• Length of minor axis = 2b = 12

• Eccentricity, e = c/a = √13/7

• Length of latus rectum = 2b²/a = (2 × 36)/7 = 72/7

Ans: The denominator of y²/400 is greater than that of x²/100, so the major axis is along the y-axis.

Comparing with x²/b² + y²/a² = 1, we get a = 20 and b = 10.

Now, c = √(a² − b²) = √(400 − 100) = √300 = 10√3

• Coordinates of the foci: (0, 10√3) and (0, −10√3)

• Coordinates of the vertices: (0, 20) and (0, −20)

• Length of major axis = 2a = 40

• Length of minor axis = 2b = 20

• Eccentricity, e = c/a = 10√3/20 = √3/2

• Length of latus rectum = 2b²/a = (2 × 100)/20 = 10

Ans: Dividing the entire equation by 144, we get:

36x²/144 + 4y²/144 = 1, which gives x²/4 + y²/36 = 1

The denominator of y²/36 is greater than that of x²/4, so the major axis is along the y-axis.

Comparing with x²/b² + y²/a² = 1, we get a = 6 and b = 2.

Now, c = √(a² − b²) = √(36 − 4) = √32 = 4√2

• Coordinates of the foci: (0, 4√2) and (0, −4√2)

• Coordinates of the vertices: (0, 6) and (0, −6)

• Length of major axis = 2a = 12

• Length of minor axis = 2b = 4

• Eccentricity, e = c/a = 4√2/6 = 2√2/3

• Length of latus rectum = 2b²/a = (2 × 4)/6 = 4/3

Ans: Dividing the entire equation by 16, we get:

x²/1 + y²/16 = 1

The denominator of y²/16 is greater than that of x²/1, so the major axis is along the y-axis.

Comparing with x²/b² + y²/a² = 1, we get a = 4 and b = 1.

Now, c = √(a² − b²) = √(16 − 1) = √15

• Coordinates of the foci: (0, √15) and (0, −√15)

• Coordinates of the vertices: (0, 4) and (0, −4)

• Length of major axis = 2a = 8

• Length of minor axis = 2b = 2

• Eccentricity, e = c/a = √15/4

• Length of latus rectum = 2b²/a = (2 × 1)/4 = ½

Ans: Dividing the entire equation by 36, we get:

x²/9 + y²/4 = 1

The denominator of x²/9 is greater than that of y²/4, so the major axis is along the x-axis.

Comparing with x²/a² + y²/b² = 1, we get a = 3 and b = 2.

Now, c = √(a² − b²) = √(9 − 4) = √5

• Coordinates of the foci: (√5, 0) and (−√5, 0)

• Coordinates of the vertices: (3, 0) and (−3, 0)

• Length of major axis = 2a = 6

• Length of minor axis = 2b = 4

• Eccentricity, e = c/a = √5/3

• Length of latus rectum = 2b²/a = (2 × 4)/3 = 8/3

In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions:

Ans: Since the vertices are on the x-axis, the major axis is along the x-axis, and the equation of the ellipse is of the form x²/a² + y²/b² = 1.

From the vertices (±5, 0), we get a = 5. From the foci (±4, 0), we get c = 4.

Using the relation c² = a² − b²: 16 = 25 − b² b² = 25 − 16 = 9

Therefore, the equation of the ellipse is x²/25 + y²/9 = 1