If y = mx + 1 is tangent to the parabola y `= 2 sqrt(x)` , then find the value of m

Answer

To find the value of \( m \) such that the line \( y = mx + 1 \) is tangent to the parabola \( y = 2\sqrt{x} \), we can follow these steps:

### Step 1: Write the equations
The equation of the parabola is given by:
\[ y = 2\sqrt{x} \]
The equation of the line is:
\[ y = mx + 1 \]

### Step 2: Set the equations equal to each other
To find the points of intersection, we set the two equations equal:
\[ mx + 1 = 2\sqrt{x} \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ 2\sqrt{x} - mx - 1 = 0 \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides to eliminate the square root leads to:
\[ (2\sqrt{x})^2 = (mx + 1)^2 \]
This simplifies to:
\[ 4x = m^2x^2 + 2mx + 1 \]

### Step 5: Rearrange into standard quadratic form
Rearranging gives us:
\[ m^2x^2 + (2m - 4)x + 1 = 0 \]

### Step 6: Use the condition for tangency
For the line to be tangent to the parabola, the quadratic equation must have exactly one solution. This occurs when the discriminant is zero:
\[ D = b^2 - 4ac = 0 \]
Here, \( a = m^2 \), \( b = 2m - 4 \), and \( c = 1 \).

### Step 7: Calculate the discriminant
The discriminant is:
\[ (2m - 4)^2 - 4(m^2)(1) = 0 \]
Expanding this gives:
\[ 4m^2 - 16m + 16 - 4m^2 = 0 \]
This simplifies to:
\[ -16m + 16 = 0 \]

### Step 8: Solve for \( m \)
Solving for \( m \):
\[ 16m = 16 \implies m = 1 \]

Thus, the value of \( m \) is:
\[ \boxed{1} \]

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