A common tangent is drawn to the circle `x^2+y^2=a^2` and the parabola `y^2=4bx`. If the angle which his tangent makes with the axis of x is `pi/4` , then the relationship between a and b (a,b`gt`0)

Options

A. A. `b=sqrt2a`
B. B. `a=bsqrt2`
C. C. c=2a
D. D. a=2c

Answer

To solve the problem, we need to find the relationship between \( a \) and \( b \) given the conditions of the common tangent to the circle and the parabola. Let's break down the solution step by step.

### Step 1: Identify the equations
The equations given are:
1. Circle: \( x^2 + y^2 = a^2 \)
2. Parabola: \( y^2 = 4bx \)

### Step 2: Determine the slope of the tangent
We know that the angle the tangent makes with the x-axis is \( \frac{\pi}{4} \). Therefore, the slope \( m \) of the tangent line is:
\[ m = \tan\left(\frac{\pi}{4}\right) = 1 \]

### Step 3: Write the equation of the tangent line
The general equation of the tangent to the parabola \( y^2 = 4bx \) at a point can be expressed as:
\[ y = mx + \frac{b}{m} \]
Substituting \( m = 1 \):
\[ y = x + b \]
Rearranging gives us:
\[ x - y + b = 0 \quad \text{(Equation 1)} \]

### Step 4: Find the perpendicular distance from the center of the circle to the tangent line
The center of the circle is at \( (0, 0) \). The formula for the perpendicular distance \( d \) from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
For our tangent line \( x - y + b = 0 \), we have \( A = 1, B = -1, C = b \). Thus, the perpendicular distance from the center \( (0, 0) \) is:
\[ d = \frac{|1 \cdot 0 + (-1) \cdot 0 + b|}{\sqrt{1^2 + (-1)^2}} = \frac{|b|}{\sqrt{2}} = \frac{b}{\sqrt{2}} \quad \text{(since } b > 0\text{)} \]

### Step 5: Set the perpendicular distance equal to the radius
Since the line is a tangent to the circle, this distance must equal the radius \( a \):
\[ \frac{b}{\sqrt{2}} = a \]

### Step 6: Rearrange to find the relationship between \( a \) and \( b \)
Multiplying both sides by \( \sqrt{2} \):
\[ b = a\sqrt{2} \]

### Final Relationship
Thus, the relationship between \( a \) and \( b \) is:
\[ b = \sqrt{2} a \]