Tangent and normal are drawn at the point `P-=(16 ,16)` of the parabola `y^2=16 x` which cut the axis of the parabola at the points `A` and `B` , respectively. If the center of the circle through `P ,A ` and `B` is `C` , then the angle between `P C` and the axis of `x` is

Options

A. (a) `tan^(-1)(1/2)`
B. (b) `tan^(-1)2`
C. (c) `tan^(-1)(3/4)`
D. (d) `tan^(-1)(4/3)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the parabola and the point P
The given parabola is \( y^2 = 16x \). The coordinates of the point \( P \) are \( (16, 16) \).

### Step 2: Find the coordinates of the focus of the parabola
The standard form of the parabola is \( y^2 = 4ax \). Here, comparing with \( y^2 = 16x \), we find that \( 4a = 16 \) which gives \( a = 4 \). The focus of the parabola is at \( (a, 0) = (4, 0) \).

### Step 3: Determine the tangent and normal at point P
The equation of the tangent to the parabola at point \( P(x_1, y_1) \) is given by:
\[ yy_1 = 8(x + x_1) \]
Substituting \( P(16, 16) \):
\[ y \cdot 16 = 8(x + 16) \]
This simplifies to:
\[ 16y = 8x + 128 \implies 2y = x + 16 \implies x - 2y + 16 = 0 \]
This is the equation of the tangent line.

The equation of the normal at point \( P \) is given by:
\[ y - y_1 = -\frac{1}{m}(x - x_1) \]
where \( m \) is the slope of the tangent. The slope \( m \) at point \( P \) can be calculated from the derivative of the parabola.

The derivative of \( y^2 = 16x \) gives:
\[ 2y \frac{dy}{dx} = 16 \implies \frac{dy}{dx} = \frac{8}{y} = \frac{8}{16} = \frac{1}{2} \text{ at } P(16, 16) \]
Thus, the slope of the normal is \( -2 \). The equation of the normal line is:
\[ y - 16 = -2(x - 16) \implies y - 16 = -2x + 32 \implies 2x + y - 48 = 0 \]

### Step 4: Find points A and B where tangent and normal intersect the x-axis
To find point \( A \) (where the tangent intersects the x-axis), set \( y = 0 \) in the tangent equation:
\[ x - 2(0) + 16 = 0 \implies x = -16 \]
So, \( A(-16, 0) \).

To find point \( B \) (where the normal intersects the x-axis), set \( y = 0 \) in the normal equation:
\[ 2x + 0 - 48 = 0 \implies 2x = 48 \implies x = 24 \]
So, \( B(24, 0) \).

### Step 5: Find the center of the circle through points P, A, and B
The center \( C \) of the circle passing through points \( P(16, 16) \), \( A(-16, 0) \), and \( B(24, 0) \) is located at the focus of the parabola, which we found earlier to be \( C(4, 0) \).

### Step 6: Calculate the angle between line PC and the x-axis
To find the angle \( \alpha \) between line \( PC \) and the x-axis, we need the slope of line \( PC \):
\[ \text{slope of } PC = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 16}{4 - 16} = \frac{-16}{-12} = \frac{4}{3} \]
The angle \( \alpha \) can be found using:
\[ \tan(\alpha) = \frac{4}{3} \]
Thus, the angle \( \alpha \) can be calculated as:
\[ \alpha = \tan^{-1}\left(\frac{4}{3}\right) \]

### Final Answer
The angle between \( PC \) and the x-axis is \( \tan^{-1}\left(\frac{4}{3}\right) \).

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