Question 37 of 70beginner🔧 ApplyShort Answer2 marks

In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse: x²/16 + y²/9 = 1

Correct Answer

Given equation is x²/16 + y²/9 = 1. Since the denominator of x²/16 (which is 16) is larger than the denominator of y²/9 (which is 9), the major axis is along the x-axis.

Comparing with the standard equation x²/a² + y²/b² = 1, we have a² = 16 and b² = 9. Therefore, a = 4 and b = 3.

c = a2−b2\sqrt{a² - b²} = 16−9\sqrt{16 - 9} = 7\sqrt{7}

Eccentricity (e) = c/a = 7\sqrt{7}/4

Foci: (±7\sqrt{7}, 0)

Vertices: (±4, 0)

Length of major axis = 2a = 8 units

Length of minor axis = 2b = 6 units

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

Exercise: EXERCISE 10.3 | Q: 3 | (Chapter: 20)
For More Understanding

Explanation

Following the method shown in Example 9 from the textbook context, we first identify which denominator is larger to determine the major axis direction. Since 16 > 9, the major axis is along the x-axis. We then extract a² = 16 and b² = 9 from the standard form. The value of c is calculated using c = a2−b2\sqrt{a² - b²}, and all other properties follow from the standard formulas for ellipses.

Solution Steps

  1. Step 1: Identify that denominator of x²/16 (i.e., 16) is larger than denominator of y²/9 (i.e., 9), so major axis is along x-axis.

  2. Step 2: Compare with standard form x²/a² + y²/b² = 1 to get a² = 16, b² = 9, hence a = 4, b = 3.

  3. Step 3: Calculate c = a2−b2\sqrt{a² - b²} = 16−9\sqrt{16 - 9} = 7\sqrt{7}.

  4. Step 4: Find eccentricity e = c/a = 7\sqrt{7}/4.

  5. Step 5: Determine foci (±c, 0) = (±7\sqrt{7}, 0) and vertices (±a, 0) = (±4, 0).

  6. Step 6: Calculate length of major axis = 2a = 8 units, minor axis = 2b = 6 units, and latus rectum = 2b²/a = 9/2 units.

PreviousNext