Question

The line $y = mx + 2$ is a tangent to the parabola $y^{2} = 8x$ if}

• A. $m = 1$
• B. $m = 2$
• C. $m = 3$
• D. $m = 4$

Hint:

Remember $c = a/m$ for $y^2 = 4ax$. It’s a shortcut that avoids using the discriminant method ($\Delta = 0$).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The condition for a line \(y = mx + c\) to be a tangent to the parabola \(y^2 = 4ax\) is \(c = \frac{a}{m}\).
Step 2: Key Formula or Approach:
1. Identify \(a\) from the parabola equation \(y^2 = 4ax\).
2. Apply the tangency condition: \(c = \frac{a}{m}\).
Step 3: Detailed Explanation:
The given parabola is \( y^2 = 8x \). Comparing with \( y^2 = 4ax \): \[ 4a = 8 \implies a = 2 \] The given line is \( y = mx + 2 \). Comparing with \( y = mx + c \): \[ c = 2 \] Applying the tangency condition \( c = \frac{a}{m} \): \[ 2 = \frac{2}{m} \] \[ 2m = 2 \implies m = 1 \]
Step 4: Final Answer:
The value of \( m \) is 1.