Tangents are drawn to hyperbola `(x^2)/(16)-(y^2)/(b^2)=1`. (b being parameter) from A(0, 4). The locus of the point of contact of these tangent is a conic C, then:

Options

A. Eccentricity of conic C is 1
B. (0, 3) is focus of C
C. Eccentricity of conic C is `(1)/(2)`
D. (0, 5) is focus of C

Answer

To solve the problem step by step, we will find the locus of the point of contact of the tangents drawn from the point A(0, 4) to the hyperbola given by the equation \(\frac{x^2}{16} - \frac{y^2}{b^2} = 1\).

### Step 1: Identify the hyperbola and the point of tangency
The hyperbola is given by the equation:
\[ \frac{x^2}{16} - \frac{y^2}{b^2} = 1 \]
The point from which tangents are drawn is \(A(0, 4)\).

### Step 2: Write the equation of the chord of contact
The chord of contact from point \(A(x_1, y_1)\) to the hyperbola is given by:
\[ \frac{x_1 x}{16} - \frac{y_1 y}{b^2} = 1 \]
Substituting \(A(0, 4)\) into the equation:
\[ \frac{0 \cdot x}{16} - \frac{4y}{b^2} = 1 \]
This simplifies to:
\[ -\frac{4y}{b^2} = 1 \quad \Rightarrow \quad 4y = -b^2 \quad \Rightarrow \quad y = -\frac{b^2}{4} \]

### Step 3: Substitute \(b^2\) in terms of \(y\)
From the previous step, we have \(b^2 = -4y\). Now, substitute this into the hyperbola equation:
\[ \frac{x^2}{16} - \frac{y^2}{-4y} = 1 \]
This simplifies to:
\[ \frac{x^2}{16} + \frac{y^2}{4y} = 1 \]

### Step 4: Rearranging the equation
To eliminate the fraction, multiply through by \(16y\):
\[ x^2y + 4y^2 = 16y \]
Rearranging gives:
\[ x^2y + 4y^2 - 16y = 0 \]

### Step 5: Finding the locus
This equation represents the locus of the point of contact of the tangents. To analyze this conic, we can rewrite it in a more standard form.

Factoring out \(y\):
\[ y(x^2 + 4y - 16) = 0 \]
This gives us \(y = 0\) or \(x^2 + 4y - 16 = 0\).

### Step 6: Solve for \(y\)
From \(x^2 + 4y - 16 = 0\):
\[ 4y = 16 - x^2 \quad \Rightarrow \quad y = 4 - \frac{x^2}{4} \]
This is the equation of a downward-opening parabola.

### Conclusion
The locus of the point of contact of the tangents is a parabola given by:
\[ y = 4 - \frac{x^2}{4} \]
The eccentricity of a parabola is always \(1\), and the focus can be found by comparing it to the standard form of a parabola.