Question

The equation of the common tangents to the parabola y = x² and y = –(x – 2)² is / are 

(a)   y = 4(x–1)

(b)   y = 0 

(c)  y = –4(x–1) 

(d)  y = –30x–50 

Answer 1

Correct option (a),(b)

Explanation:

Let y = mx + c is tangent to y = x²

mx + c = x² ⇒  x²– mx – c = 0 has equal roots

m² + 4c = 0

y = mx – m²/4 is tangent to y = –(x–2)²

also mx – 4 m² = –x²  + 4x – 4

x²  +(m – 4)x + 4 - m²/4 =  has equal roots

(m – 4)² - 4(4 - m²/4) = 0

m² + 16 – 8m – 16 + m² = 0

m² – 4m = 0

m = 0, 4

Equation of tangent are y = 0 and y = 4x – 4