The straight line x + y =a will be a tangent to the ellipse `x^2//9+y^2//16=1` if a=

Options

A. 8
B. `pm5`
C. `pm10`
D. `pm6`

Answer

To determine the value of \( a \) for which the line \( x + y = a \) is a tangent to the ellipse given by the equation \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \), we can follow these steps:

### Step 1: Identify the slope of the line
The line can be rewritten as:
\[ y = -x + a \]
From this, we can see that the slope \( m \) of the line is \( -1 \).

### Step 2: Identify the parameters of the ellipse
The equation of the ellipse is given as:
\[ \frac{x^2}{9} + \frac{y^2}{16} = 1 \]
Here, we can identify \( a^2 = 9 \) and \( b^2 = 16 \). Therefore, we have:
\[ a = 3 \quad \text{and} \quad b = 4 \]

### Step 3: Use the tangent line formula
The formula for the equation of the tangent line to the ellipse at a point can be expressed as:
\[ y = mx + \sqrt{a^2 m^2 + b^2} \]
Substituting \( m = -1 \), \( a = 3 \), and \( b = 4 \) into the formula gives:
\[ y = -x + \sqrt{3^2(-1)^2 + 4^2} \]
Calculating the expression under the square root:
\[ \sqrt{9 \cdot 1 + 16} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus, the equation of the tangent line becomes:
\[ y = -x \pm 5 \]

### Step 4: Write the equations of the tangents
This results in two equations:
1. \( y = -x + 5 \)
2. \( y = -x - 5 \)

### Step 5: Compare with the given line
We need to compare these equations with the given line \( y = -x + a \). This gives us two possible values for \( a \):
\[ a = 5 \quad \text{or} \quad a = -5 \]

### Conclusion
Thus, the values of \( a \) for which the line \( x + y = a \) is a tangent to the ellipse are:
\[ a = 5 \quad \text{or} \quad a = -5 \]