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The circle above is defined by the equation: x 2 + y 2 = 100. A tangent line (4x – 3y = 50) intersects the circle at a point of tangency: (8,6). The tangent.

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Presentation on theme: "The circle above is defined by the equation: x 2 + y 2 = 100. A tangent line (4x – 3y = 50) intersects the circle at a point of tangency: (8,6). The tangent."— Presentation transcript:

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2 The circle above is defined by the equation: x 2 + y 2 = 100. A tangent line (4x – 3y = 50) intersects the circle at a point of tangency: (8,6). The tangent line is perpendicular to the radius of the circle. The circle above is defined by the equation: x 2 + y 2 = 100. A tangent line (4x – 3y = 50) intersects the circle at a point of tangency: (8,6). The tangent line is perpendicular to the radius of the circle. 4x – 3y = 50 tangent line (8,6) point of tangency The slope of the radius is the negative reciprocal to the slope of the tangent line. x 2 + y 2 = 100 circle The circle above is defined by the equation: x 2 + y 2 = 100. A tangent line (4x – 3y = 50) intersects the circle at a point of tangency: (8,6). The tangent line is perpendicular to the radius of the circle. The circle above is defined by the equation: x 2 + y 2 = 100. A tangent line (4x – 3y = 50) intersects the circle at a point of tangency: (8,6). The tangent line is perpendicular to the radius of the circle. m r = m t =

3 When a tangent and a radius intersect at the point of tangency, they are always perpendicular to each other. It then follows that their slopes are always negative reciprocals of each other. C(-1,2) P(2,-1) T(0,-3) P(2,-1) m r =

4 Find the slope of the line tangent to the circle x 2 + y 2 = 5 and passing through the point R(-2,1). R(-2,1)

5 Find the equation of the tangent to the circle x 2 + y x – 24y = 0 and passing through the point T(0,0). Step 1: Find the centre and the radius. x 2 + y x – 24y = 0 (x x + 25) + (y 2 – 24y + 144) = (x + 5) 2 + (y – 12) 2 = 169 Centre: (-5,12)r = 13 12(y – 0) = 5(x – 0) 12y = 5x 5x – 12y = 0 Step 2: Find the slope of the radius. Step 3: Find the slope of the tangent. Step 4: Find the equation of the tangent.

6 Find the equation of the tangent to the circle x 2 + y 2 – 6y - 16 = 0 and passing through the point T(3,7). Step 1: Find the centre and the radius. x 2 + y 2 – 6y - 16 = 0 x 2 + (y 2 – 6y + 9) = x 2 + (y – 3) 2 = 25 Centre: (0,3) r = 5 4(y – 7) = -3(x – 3) 4y - 28 = -3x + 9 3x + 4y = x + 4y = 37

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