InterviewSolution
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InterviewSolution
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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 |
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Answer» `(1)/(SQRT(2))` `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 ory = - 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)` |
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