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What is the equation of the line tangent to x^2 + y^2 = 25 at (3, 4) (without the use...

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user1688 | eNotes Newbie

Posted July 2, 2013 at 3:53 AM via web

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What is the equation of the line tangent to x^2 + y^2 = 25 at (3, 4) (without the use of calculus)

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted July 2, 2013 at 4:00 AM (Answer #1)

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The equation of the tangent to the circle represented by x^2 + y^2 = 25 at the point (3, 4) has to be determined.

At any point on a circle, the tangent is perpendicular to the radial line at that point. For x^2 + y^2 = 25, the center is (0,0), the slope of the radial line from (3, 4) to (0,0) is `(4 - 0)/(3 - 0) = 4/3` . If a line has slope `s` , the slope of a perpendicular line is `-1/s` . This gives the slope of the tangent as `-3/4` . The equation of the tangent is `(y - 4)/(x - 3) = -3/4`

=> 4(y - 4) = 3(3 - x)

=> 4y - 16 = 9 - 3x

=> 3x + 4y - 25 = 0

The required equation of the tangent to the circle x^2 + y^2 = 25 at the point (3, 4) is 3x + 4y - 25 = 0

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