Question

The equation of a common tangent to the parabolas \( y = x^2 \) and \( y = -(x - 2)^2 \) is:

• A. \( y = 4(x - 2) \)
• B. \( y = 4(x - 1) \)
• C. \( y = 4(x + 1) \)
• D. \( y = 4(x + 2) \)

Hint:

When finding the common tangent to two curves, use the discriminant condition for tangency and solve the resulting system of equations to determine the parameters of the tangent line.

Answer 1

The Correct Option is B

Equation of tangent of parabola \( y = x^2 \) be \[ tx = y + at^2 \quad \dots (i) \] \[ y = tx - \frac{t^2}{4} \] Solve with \( y = -(x - 2)^2 \): \[ tx - \frac{t^2}{4} = -(x - 2)^2 \] \[ x^2 + x(t - 4) - \frac{t^2}{4} + 4 = 0 \] Here, Discriminant \( = 0 \). \[ (t - 4)^2 - 4 \cdot \left( 4 - \frac{t^2}{4} \right) = 0 \] \[ \Rightarrow t^2 - 4t = 0 \Rightarrow t = 0 \quad {or} \quad t = 4 \] Put value of \( t \) in eq. (i), then \[ y = 4(x - 1). \]