Question

Answer the following:

Show that the line 2x − y = 4 touches the hyperbola 4x² − 3y² = 24. Find the point of contact

Answer 1

Given equation of the hyperbola is 4x² − 3y² = 24.

∴ `x^2/6 - y^2/8` = 1

Comparing this equation with `x^2/"a"^2 - y^2/"b"^2` = 1, we get

a² = 6 and b² = 8

Given equation of line is 2x – y = 4

∴ y = 2x – 4

Comparing this equation with y = mx + c, we get

m = 2 and c = – 4

For the line y = mx + c to be a tangent to the hyperbola `x^2/"a"^2 - y^2/"b"^2` = 1, we must have

c² = a²m² – b²

c² = (– 4)² = 16

a²m² – b² = 6(2)² – 8

= 24 – 8

= 16

∴ The given line is a tangent to the given hyperbola and point of contact

= `(- ("a"^2"m")/"c", - "b"^2/"c")`

= `((-6(2))/(-4), (-8)/(-4))`

= (3, 2).