For the ellipse `(x^(2))/(25)+(y^(2))/(16) = 1`, a list of lines given in List-I are to be matched with their equation given in list II
`{:(list I,list II),("directrix corresponding to the focus" (-3,0), y=4),("tangent at the vertex" (0,4),3x=25),("latus rectum through" (3,0),x=3),("" ,y+4=0 ),("" ,x+3=0),("" ,3x+25=0):}`

Options

A. b a e
B. f a c
C. b d c
D. f a e

Answer

To solve the problem, we need to analyze the given ellipse and match the lines from List-I with their corresponding equations in List-II.

### Step 1: Identify the parameters of the ellipse

The given equation of the ellipse is:

\[ \frac{x^2}{25} + \frac{y^2}{16} = 1 \]

From this equation, we can identify:
- \( a^2 = 25 \) (thus \( a = 5 \))
- \( b^2 = 16 \) (thus \( b = 4 \))

### Step 2: Calculate the value of \( c \)

To find the distance of the foci from the center, we use the formula:

\[ c^2 = a^2 - b^2 \]

Calculating \( c \):

\[ c^2 = 25 - 16 = 9 \quad \Rightarrow \quad c = 3 \]

The foci of the ellipse are at the points \( (c, 0) \) and \( (-c, 0) \), which gives us the foci at \( (3, 0) \) and \( (-3, 0) \).

### Step 3: Determine the equations for the lines in List-I

1. **Directrix corresponding to the focus (-3, 0)**:
- The directrix for an ellipse is given by \( x = -\frac{a}{e} \), where \( e = \frac{c}{a} \).
- Here, \( e = \frac{3}{5} \).
- Thus, the directrix is \( x = -\frac{5}{\frac{3}{5}} = -\frac{25}{3} \).
- The equation of the directrix is \( 3x + 25 = 0 \).

2. **Tangent at the vertex (0, 4)**:
- The vertex at \( (0, 4) \) corresponds to the maximum value of \( y \).
- The tangent line at this vertex is simply \( y = 4 \).

3. **Latus rectum through (3, 0)**:
- The latus rectum is a vertical line through a focus.
- The equation is \( x = 3 \).

### Step 4: Match the lines from List-I with List-II

Now we can match the lines from List-I with their corresponding equations in List-II:

- **Directrix corresponding to the focus (-3, 0)**: Matches with \( 3x + 25 = 0 \) (last option in List-II).
- **Tangent at the vertex (0, 4)**: Matches with \( y = 4 \) (first option in List-II).
- **Latus rectum through (3, 0)**: Matches with \( x = 3 \) (third option in List-II).

### Final Matching

- **Directrix**: \( 3x + 25 = 0 \) (matches with last option).
- **Tangent at the vertex**: \( y = 4 \) (matches with first option).
- **Latus rectum**: \( x = 3 \) (matches with third option).

### Summary of Matches

- Directrix corresponding to the focus (-3, 0) → \( 3x + 25 = 0 \) (last option)
- Tangent at the vertex (0, 4) → \( y = 4 \) (first option)
- Latus rectum through (3, 0) → \( x = 3 \) (third option)