If the line ax + y = c, touchs both the curves `x^(2) + y^(2) = 1`
and `y^(2) = 4 sqrt(2)x`, then |c| is equal to

Options

A. `(1)/(sqrt(2))`
B. 2
C. `sqrt(2)`
D. `(1)/(2)`

Answer

C

Explanation

Use the equation of tangent of slope 'm' to the parabola `y^(2) = 4 ax` is `y = mx + (a)/(m)` and a line ax + by + c = 0 touch the circle
`x^(2) + y^(2) = r^(2)`, if `(|c|)/(sqrt(a^(2) + b^(2)) = r`
Since equation of given parabola is `y^(2) = 4 sqrt(2x)` and equation of tangent line is ax + y = c or y = - ax + c
then `c = (sqrt(2))/(m) = (sqrt(2))/(-a)` [`:.` m = slope of line = - a]
[`:'` line y = mx + c touches the parabola
`y^(2) = 4ax` if c = a/m]
Then, equation of tangent line becomes
`y = - ax - (sqrt(2))/(a)` ......(i)
`:.` Line (i) is also tangent to the circle `x^(2) + y^(2) = 1`
`:.` Radius = 1 = `(|-(sqrt(2))/(a)|)/(sqrt(1 + a^(2)) implies sqrt(1 + a^(2)) = | - (sqrt(2))/(a)|)`
`implies 1 + a^(2) = (2)/(a^(2))` [squaring both side]
`implies a^(4) + a^(2) - 2 = 0 implies (a^(2) + 2) (a^(2) - 1) = 0`
`implies a^(2) = 1` `[:. a^(2) gt 0, AA a in R]`
`:. |c| = (sqrt(2))/(|a|) = sqrt(2)`