Question

If the line y = x + k  touches the hyperbola 9x² -16y² = 144, then k = _________

Options

• 7

• -7

• `+-sqrt7`

• `+-sqrt19`

Answer 1

If the line y = x + k  touches the hyperbola 9x² -16y² = 144, then k = `+-sqrt7`

The given line is
y = x + k   ....(i)
Comparing with y = mx + c, we get
m = 1
c = k
The given hyperbola is
9x²−16y² = 144 ....(ii)
⇒ `((9x^2)/144) − ((16y^2)/144) =1` [Dividing throughout by 144]
⇒ `x^2/16 ​− y^2/9 ​= 1`
Comparing with `(x^2)/(a^2) ​− (y^2)/(b^2) ​= 1`, we get
a² = 16
and b² = 9
If line (i) touches the hyperbola (ii), then
c² = a²m² − b²
⇒ k² = (16)(1)² − 9
⇒ k² = 16 − 9
⇒ k² = 7
⇒ k = `+-sqrt7​`