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What is value of a if ax + 2y = 9 is tangent to the circle x^2 + y^2 = 16  

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carolinesmiths | Student, Kindergarten | (Level 1) Salutatorian

Posted July 3, 2012 at 2:16 AM via web

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What is value of a if ax + 2y = 9 is tangent to the circle x^2 + y^2 = 16

 

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted July 3, 2012 at 2:34 AM (Answer #1)

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The equation of the circle x^2 + y^2 = 16 is (x - 0)^2 + (y - 0)^2 = 4^2 which has a center (0, 0) and a radius 4. If the line ax + 2y = 9 is a tangent to the circle, the perpendicular distance of the line from the center of the circle is 4.

As the perpendicular distance of a line ax + by + c = 0 from a point (X, Y) is equal to `D = |aX + bY + c|/sqrt(a^2 + b^2)` and the line ax + 2y + 9 = 0 has a perpendicular distance of 4 from (0,0) this gives:

 `4 = |a*0 + b*0 - 9|/sqrt(a^2 + 4)`

=> `sqrt(a^2 + 4) = 9/4`

=> `a^2 + 4 = 81/16`

=> `a^2 = 17/16`

=> `a = sqrt 17 / 4` and `a = -sqrt 17/4`

The required value of a can be `a = sqrt 17/4` and `a = -sqrt 17/4`

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