Question

If the line \( y = kx \) touches the parabola \( y = (x-1)^2 \), then the values of \( k \) are

• A. \( 2, -2 \)
• B. \( 0, 4 \)
• C. \( 0, -2 \)
• D. \( 0, 2 \)
• E. \( 0, -4 \)

Hint:

For tangency, always reduce to quadratic and set discriminant = 0.

Answer 1

The Correct Option is B

Concept:
• A line touches a parabola if it intersects at exactly one point.
• This happens when the quadratic equation formed has equal roots ⇒ discriminant = 0.

Step 1: Equate line and parabola.
\[ kx = (x-1)^2 \] \[ kx = x^2 - 2x + 1 \] \[ x^2 - (2+k)x + 1 = 0 \]

Step 2: Condition for tangency.
For one point of intersection: \[ D = 0 \] \[ (2+k)^2 - 4(1)(1) = 0 \]

Step 3: Solve for \(k\).
\[ (2+k)^2 = 4 \] \[ 2+k = \pm 2 \] \[ k = 0 \quad \text{or} \quad k = -4 \]

Step 4: Check options.
Correct values: \[ k = 0, -4 \]

Step 5: Final Answer.
\[ \boxed{(E)} \]