If the line y = x +k touches the ellipse `9x^(2) + 16y^(2) = 144,` then
(i) k = 5 only
(ii) k = - 5 only
(iii) k = 5, - 5
(iv) none of these

Options

A. k = 5 only
B. k = - 5 only
C. k = 5, - 5
D. none of these

Answer

To solve the problem, we need to determine the values of \( k \) for which the line \( y = x + k \) is a tangent to the ellipse given by the equation \( 9x^2 + 16y^2 = 144 \).

### Step 1: Convert the ellipse equation to standard form
The given equation of the ellipse is:
\[ 9x^2 + 16y^2 = 144 \]
To convert it to standard form, we divide the entire equation by 144:
\[ \frac{9x^2}{144} + \frac{16y^2}{144} = 1 \]
This simplifies to:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]
Here, we can identify \( a^2 = 16 \) and \( b^2 = 9 \).

### Step 2: Write the general equation of the tangent to the ellipse
The general equation of the tangent to the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is given by:
\[ y = mx + \sqrt{a^2m^2 + b^2} \]
where \( m \) is the slope of the tangent line.

### Step 3: Identify the slope from the given line
From the equation of the line \( y = x + k \), we see that the slope \( m = 1 \).

### Step 4: Substitute \( m \) into the tangent equation
Substituting \( m = 1 \), \( a^2 = 16 \), and \( b^2 = 9 \) into the tangent equation:
\[ y = x + \sqrt{16(1)^2 + 9} \]
This simplifies to:
\[ y = x + \sqrt{16 + 9} = x + \sqrt{25} = x + 5 \]

### Step 5: Compare with the given line equation
Now, we compare this with the line \( y = x + k \):
\[ k = 5 \]

### Step 6: Consider the negative root
Since the tangent can also be below the ellipse, we consider the negative root:
\[ k = -5 \]

### Conclusion
Thus, the possible values of \( k \) are \( 5 \) and \( -5 \).

The final answer is:
\[ \text{(iii) } k = 5, -5 \]

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