Question

The line \(y = mx + 3\) is tangent to the parabola \(y^2 = 4x\), if the value of m is

• A. \(3\)
• B. \(1/3\)
• C. \(4\)
• D. \(1/4\)

Hint:

If a variable line is tangent to a conic, substitution followed by the condition “discriminant = 0” is usually the quickest method.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For a line to touch a parabola, their equations must have only one common solution. This leads to a standard condition on the constants.
Step 2: Key Formula or Approach:
The line \(y = mx + c\) is tangent to \(y^2 = 4ax\) if \(c = \frac{a}{m}\).
Step 3: Detailed Explanation:
1. Identify \(a\) from the parabola: \(y^2 = 4x \implies 4a = 4 \implies a = 1\).
2. Identify \(c\) from the line: \(y = mx + 3 \implies c = 3\).
3. Use the tangency condition:
\[ 3 = \frac{1}{m} \] \[ m = \frac{1}{3} \] Step 4: Final Answer:
The value of \(m\) is \(\frac{1}{3}\).