Question

The equation of the tangent to the parabola \(y^2=8x\) at point \((2,4)\) is:

• A. \(y=x+2\)
• B. \(y=2x+2\)
• C. \(y=x+4\)
• D. \(y=2x\)

Hint:

Memorize standard tangent forms for conics.

Answer 1

The Correct Option is B

To find the equation of the tangent to the parabola \( y^2 = 8x \) at the point \( (2,4) \), we can use the concept of the derivative of the equation to find the slope of the tangent at that point.

1. The equation of the given parabola is \( y^2 = 8x \).
2. Differentiate the implicit equation with respect to \( x \) to find the slope of the tangent. 
   Differentiating both sides: \[ \frac{d}{dx}(y^2) = \frac{d}{dx}(8x) \] This gives us: \[ 2y \frac{dy}{dx} = 8 \]
3. Solving for \( \frac{dy}{dx} \), the derivative: 
   \[ \frac{dy}{dx} = \frac{8}{2y} = \frac{4}{y} \]
4. Substitute the coordinates of the point \( (2,4) \) into the derivative to find the slope of the tangent line at that point: 
   \[ \frac{dy}{dx} = \frac{4}{4} = 1 \]
5. Use the point-slope form of the line to find the equation of the tangent: 
   \[ y - y_1 = m(x - x_1) \] Where \( (x_1, y_1) = (2, 4) \) and \( m = 1 \). 
   This gives: 
   \[ y - 4 = 1(x - 2) \]

The correct answer is \( y = 2x + 2 \).