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**Equation of Tangent line**

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**is the negative reciprocal to the slope of the tangent line.**

x2 + y2 = 100 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. mr = mt = The circle above is defined by the equation: x2 + y2 = 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: x2 + y2 = 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: x2 + y2 = 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: x2 + y2 = 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.

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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. T(0,-3) P(2,-1) C(-1,2) P(2,-1) mr =

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**Find the slope of the line tangent to the circle x2 + y2 = 5 and passing through the point R(-2,1).**

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Find the equation of the tangent to the circle x2 + y2 + 10x – 24y = 0 and passing through the point T(0,0). Step 1: Find the centre and the radius. x2 + y2 + 10x – 24y = 0 (x2 + 10x + 25) + (y2 – 24y + 144) = (x + 5)2 + (y – 12)2 = 169 Step 3: Find the slope of the tangent. Centre: (-5,12) r = 13 Step 2: Find the slope of the radius. Step 4: Find the equation of the tangent. 12(y – 0) = 5(x – 0) 12y = 5x 5x – 12y = 0

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Find the equation of the tangent to the circle x2 + y2 – 6y - 16 = 0 and passing through the point T(3,7). Step 1: Find the centre and the radius. x2 + y2 – 6y - 16 = 0 x2 + (y2 – 6y + 9) = x2 + (y – 3)2 = 25 Centre: (0,3) r = 5 4(y – 7) = -3(x – 3) 4y - 28 = -3x + 9 3x + 4y = 3x + 4y = 37

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10-5 T ANGENTS Ms. Andrejko. V OCABULARY Tangent- Line in the same plane as a circle that intersects the circle at exactly one point called the point.

10-5 T ANGENTS Ms. Andrejko. V OCABULARY Tangent- Line in the same plane as a circle that intersects the circle at exactly one point called the point.

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