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

Concept:
A line is tangent to a parabola if it intersects the parabola at exactly one point. So after substituting the line into the parabola, the resulting quadratic must have discriminant zero. ip

Step 1: Substitute the line in the parabola.
Given: \[ y=mx+3 \] From this, \[ x=\frac{y-3}{m} \] Now substitute in \[ y^2=4x \] \[ y^2 = 4\left(\frac{y-3}{m}\right) \] \[ my^2 - 4y + 12 = 0 \] ip

Step 2: Use the tangency condition.
For tangency, discriminant must be zero: \[ (-4)^2 - 4(m)(12)=0 \] \[ 16 - 48m = 0 \] \[ 48m = 16 \] \[ m=\frac{1}{3} \] ip Hence, the correct answer is:
\[ \boxed{(B)\ \frac{1}{3}} \]