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Question
Answer the following:
Show that the line 2x − y = 4 touches the hyperbola 4x2 − 3y2 = 24. Find the point of contact
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Solution
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).
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