If e is the eccentricity, l is the semi latus-rectum and S. `S^(1)` are foci of the hyperbola `9x^(2)-16y^(2)+72x-32y=16` then ascending order of l, e `SS^(1)` is

Options

A. `1,e,SS^(1)`
B. `e,1,SS^(1)`
C. `e,SS^(1),l`
D. `SS^(1),l,e`

Answer

To solve the given problem, we need to analyze the hyperbola \(9x^2 - 16y^2 + 72x - 32y = 16\) and find the eccentricity \(e\), the semi-latus rectum \(l\), and the distance between the foci \(SS'\). Finally, we will arrange these values in ascending order.

### Step 1: Rewrite the Hyperbola in Standard Form

We start with the equation of the hyperbola:
\[ 9x^2 - 16y^2 + 72x - 32y = 16 \]

First, we rearrange the equation:
\[ 9(x^2 + 8x) - 16(y^2 + 2y) = 16 \]

Next, we complete the square for the \(x\) and \(y\) terms.

For \(x^2 + 8x\):
\[ x^2 + 8x = (x + 4)^2 - 16 \]

For \(y^2 + 2y\):
\[ y^2 + 2y = (y + 1)^2 - 1 \]

Substituting these back into the equation gives:
\[ 9((x + 4)^2 - 16) - 16((y + 1)^2 - 1) = 16 \]
\[ 9(x + 4)^2 - 144 - 16(y + 1)^2 + 16 = 16 \]
\[ 9(x + 4)^2 - 16(y + 1)^2 - 128 = 16 \]
\[ 9(x + 4)^2 - 16(y + 1)^2 = 144 \]

Dividing through by 144:
\[ \frac{(x + 4)^2}{16} - \frac{(y + 1)^2}{9} = 1 \]

This is the standard form of the hyperbola.

### Step 2: Identify Parameters

From the standard form \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\):
- \(a^2 = 16 \Rightarrow a = 4\)
- \(b^2 = 9 \Rightarrow b = 3\)

### Step 3: Calculate Eccentricity \(e\)

The eccentricity \(e\) of a hyperbola is given by:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + \frac{9}{16}} = \sqrt{\frac{25}{16}} = \frac{5}{4} \]

### Step 4: Calculate Semi-latus Rectum \(l\)

The semi-latus rectum \(l\) is given by:
\[ l = \frac{b^2}{a} = \frac{9}{4} \]

### Step 5: Calculate Distance Between Foci \(SS'\)

The distance between the foci \(SS'\) is given by:
\[ SS' = 2c \quad \text{where} \quad c = \sqrt{a^2 + b^2} \]
Calculating \(c\):
\[ c = \sqrt{16 + 9} = \sqrt{25} = 5 \]
Thus,
\[ SS' = 2c = 10 \]

### Step 6: Arrange in Ascending Order

Now we have:
- \(l = \frac{9}{4} = 2.25\)
- \(e = \frac{5}{4} = 1.25\)
- \(SS' = 10\)

Arranging these in ascending order:
\[ e < l < SS' \]

### Final Answer

The ascending order of \(l\), \(e\), and \(SS'\) is:
\[ e < l < SS' \]