Question

The equation of the common tangent to the parabola x^2=108y and y^2=32x, is

Answer 1

see the solution in image.

Answer 2

it is given that the equation of the common tangent to the parabola x² = 108y and y² = 32y

equation of parabola, y² = 32y = 4(8)y

a = 8 so equation of tangent y = mx + a/m

⇒y = mx + 8/m ..........(1)

another equation of parabola is x² = 108y

⇒x² = 108(mx + 8/m)

⇒x²m= 108m²x + 108 × 8

⇒mx² - 108m²x - 864 = 0

D = b² - 4ac = 0

⇒(108m²)² - 4m(-864) = 0

⇒108m⁴ + 32m = 0

⇒27m³ + 8 = 0

⇒m = -2/3

equation of parabola is y = (-2/3)x + 8/(-2/3)

⇒y = -2x/3 - 12

⇒3y + 2x + 36 = 0

therefore, equation of common tangent is 3y + 2x + 36 = 0