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Question

The equations of the tangents to the hyperbola 4x29y2=1 which are parallel to the line 4y=5x+7 are

A
24y=30x±161
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B
30x+24y=±161
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C
30y=24±161
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D
x+y=161
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Solution

The correct option is B 24y=30x±161
Equation of hyperbola is 4x29y2=1
or x214y219=1 ...........(1) which is of the form x2a2y2b2=1
where a2=14 and b2=19
Equation of the line parallel to the line 4y=5x+7 or 5x4y+7=0 is 5x4y+λ=0
or y=54x+λ4 ...........(2)
Slope of this line=54=m(let) and c=λ4
If line(2) touches the hyperbola (1) then
c2=a2m2b2
(λ4)2=(12)2(54)2(13)2
(λ4)2=14×251619=16164×9
λ2=16136λ=±1616
Substitute the values of λ in equation(2) the required equation of tangents are
5x4y±1616=0
or 30x24y±161=0

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