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A line L varies such that length of perp...

A line `L` varies such that length of perpendicular on it from origin `O` is always 4 units. If `L` cuts x-axis and y-axis at `A` and `B` respectively then minimum value of `(OA)^(2)+(OB)^(2)` is

A

`16`

B

`32`

C

`64`

D

`128`

Text Solution

Verified by Experts

The correct Answer is:
C
Let equation of variable line be `x cos alpha + y sin alpha = 4`
`:. AS = (4 sec alpha, 0)`
`B = (0, 4 cosec alpha)`
`:. (OA)^(2)+(OB)^(2)=16(sec^(2)alpha + cosec^(2)alpha)`
`= 16(2+tan^(2)alpha+cot^(2)alpha)`
`:.` Minimum value of `(OA)^(2)+(OB)^(2)=16(2+2) =64`
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