The number of common tangents to the ellipse `(x^(2))/(16) + (y^(2))/(9) =1` and the circle `x^(2) + y^(2) = 4` is

Options

A. 2
B. 1
C. 0
D. 4

Answer

To find the number of common tangents to the ellipse \(\frac{x^2}{16} + \frac{y^2}{9} = 1\) and the circle \(x^2 + y^2 = 4\), we can follow these steps:

### Step 1: Identify the equations of the ellipse and the circle
The given ellipse is:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]
This can be rewritten in standard form where \(a^2 = 16\) and \(b^2 = 9\).

The given circle is:
\[ x^2 + y^2 = 4 \]
This can be rewritten as \(x^2 + y^2 = r^2\) where \(r^2 = 4\) implies \(r = 2\).

### Step 2: Write the equation of the tangent to the circle
The equation of the tangent to the circle at a slope \(m\) is given by:
\[ y = mx \pm \sqrt{r^2(1 + m^2)} \]
Substituting \(r^2 = 4\):
\[ y = mx \pm \sqrt{4(1 + m^2)} = mx \pm 2\sqrt{1 + m^2} \]

### Step 3: Write the equation of the tangent to the ellipse
The equation of the tangent to the ellipse at a slope \(m\) is given by:
\[ y = mx \pm \sqrt{a^2m^2 + b^2} \]
Substituting \(a^2 = 16\) and \(b^2 = 9\):
\[ y = mx \pm \sqrt{16m^2 + 9} \]

### Step 4: Set the two tangent equations equal to each other
For the tangents to be common, we set the two equations equal:
\[ mx \pm 2\sqrt{1 + m^2} = mx \pm \sqrt{16m^2 + 9} \]
We can ignore \(mx\) since it is common on both sides. This leads to two cases:
1. \(2\sqrt{1 + m^2} = \sqrt{16m^2 + 9}\)
2. \(-2\sqrt{1 + m^2} = \sqrt{16m^2 + 9}\) (This case will not yield valid solutions since the left side is negative.)

### Step 5: Solve the equation
From the first case:
\[ 2\sqrt{1 + m^2} = \sqrt{16m^2 + 9} \]
Squaring both sides:
\[ 4(1 + m^2) = 16m^2 + 9 \]
Expanding and rearranging:
\[ 4 + 4m^2 = 16m^2 + 9 \]
\[ 4 - 9 = 16m^2 - 4m^2 \]
\[ -5 = 12m^2 \]
\[ m^2 = -\frac{5}{12} \]

### Step 6: Analyze the result
Since \(m^2\) cannot be negative, this indicates that there are no real values for \(m\). Therefore, there are no common tangents to the ellipse and the circle.

### Conclusion
The number of common tangents to the ellipse and the circle is **0**.

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