Find the equation of the tangent to the circle x^2 + y^2 = 25 at (3, 4) without using calculus.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The equation of the tangent to the circle x^2 + y^2 = 25 at (3, 4) has to be determined without using calculus.

In a circle, the radius is perpendicular to the tangent at any point. Here, the equation of the circle is x^2 + y^2 = 25. The general form of the equation of a circle is (x - a)^2 + (y - b)^2 = r^2, where the center is (a, b) and the radius is r. The center of the given circle is (0,0).

The equation of the line from (0,0) to (3, 4) is y/x = 4/3.

As the tangent is perpendicular to the radius, it has a slope of -3/4.

The equation of a line with slope -3/4 and passing through (3, 4) is (y - 4)/(x - 3) = -3/4

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

=> 4y - 16 = -3x + 9

=> 3x + 4y - 25 = 0

The required equation of the radius is 3x + 4y - 25 = 0

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