A tangent and a normal are drawn at the point P `(16,16)` of the parabola `y^2 = 16x` which cut the axis of the parabola at the points A and B respectively. If the centre of the through P, A and B is C, then angle between PC and axis of x is :

Options

A. `tan^(-1)""1/2`
B. `tan^(-1)2`
C. `tan^(-1)""3/4`
D. `tan^(-1)""4/3`

Answer

To solve the problem step by step, we will follow the outlined process:

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

### Step 2: Find the slope of the tangent at point P
To find the slope of the tangent to the parabola at point \( P \), we first differentiate the equation of the parabola.

The equation \( y^2 = 16x \) can be differentiated implicitly:
\[ 2y \frac{dy}{dx} = 16 \implies \frac{dy}{dx} = \frac{16}{2y} = \frac{8}{y} \]
At point \( P(16, 16) \):
\[ \frac{dy}{dx} = \frac{8}{16} = \frac{1}{2} \]
Thus, the slope of the tangent at point \( P \) is \( \frac{1}{2} \).

### Step 3: Write the equation of the tangent line
Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) = (16, 16) \) and \( m = \frac{1}{2} \):
\[ y - 16 = \frac{1}{2}(x - 16) \]
Simplifying this:
\[ y - 16 = \frac{1}{2}x - 8 \implies y = \frac{1}{2}x + 8 \]

### Step 4: Find the x-intercept (point A)
To find point \( A \), set \( y = 0 \) in the tangent equation:
\[ 0 = \frac{1}{2}x + 8 \implies \frac{1}{2}x = -8 \implies x = -16 \]
Thus, point \( A \) is \( (-16, 0) \).

### Step 5: Find the slope of the normal at point P
The slope of the normal is the negative reciprocal of the slope of the tangent:
\[ \text{slope of normal} = -\frac{1}{\frac{1}{2}} = -2 \]

### Step 6: Write the equation of the normal line
Using the point-slope form again:
\[ y - 16 = -2(x - 16) \]
Simplifying this:
\[ y - 16 = -2x + 32 \implies y = -2x + 48 \]

### Step 7: Find the x-intercept (point B)
To find point \( B \), set \( y = 0 \) in the normal equation:
\[ 0 = -2x + 48 \implies 2x = 48 \implies x = 24 \]
Thus, point \( B \) is \( (24, 0) \).

### Step 8: 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 the focus of the parabola \( y^2 = 16x \). The focus is at \( (4, 0) \).

### Step 9: Find the angle between line PC and the x-axis
To find the angle \( \alpha \) between line \( PC \) and the x-axis, we first calculate the slope of line \( PC \):
\[ \text{slope of PC} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 16}{4 - 16} = \frac{-16}{-12} = \frac{4}{3} \]
Thus, \( \tan \alpha = \frac{4}{3} \).

### Step 10: Find the angle \( \alpha \)
Therefore, the angle \( \alpha \) is given by:
\[ \alpha = \tan^{-1}\left(\frac{4}{3}\right) \]

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

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