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Question:

If the line y=mx+7√3 is normal to the hyperbola x²/24 - y²/18 = 1, then a value of m is?

3√5

√15/2

2√5

√5/2

Solution:

x²/24 - y²/18 = 1 ⇒ a=√24; b=√18
Parametric normal: √24cosθ⋅x + √18⋅ycotθ = 42
At x=0: y = 42/√18tanθ = 7√3 (from given equation)
⇒ tanθ = √3/2 ⇒ sinθ = ±√3/√5
Slope of parametric normal = -√24cosθ/√18cotθ = m ⇒ m = -√4/3sinθ = -2/√5 or 2√5.