Worked examples 1. The P(3; 2) is a point on the circle defined by: + = x2 y2 r2 – a) Determine the equation of the circle. The equation of the circle is of the form: + = r2. x2 y2 Hence; (3)2 + (–2)2 = r2 r2 = 13 Required equation is: + = 13 x2 y2 b) Determine the equation of the tangent to The equation of the tangent through the circle at the point P(3 ; 2) P(3 ;–2) is of the type: + = xx1 yy1 r2 – + = 13 x(3) y(–2) 3x =13 Required equation is: –2y 2. Determine the equation of the tangent to the circle + = 10 at the point x2 y2 K(–3 ; 1) Method 1: (Using the formula Method 2: developed above) First determine the gradient of the radius, OK; The equation of the tangent through K is MOK = of the type: + = Mtangent = 3 (Do you recall, xx1 yy1 r2 why?) i.e: + = 10 x(–3) y(1) + = 10 Equation of tangent is: y –3x = 3x + 10 is the required equation. y y -y1 = m(x x1) – y (1) = 3(x – –(–3)) y 1 = 3x + 9 – 10 is the required equation. 3x y = + Activity 2 1. Determine the equation of the tangent to the circle at the given point. Equation of circle Given point
Worked examples 1. The P(3; – 2) is a point on the circle defined by: x2 + y2 = r2 a) Determine the equation of the circle. The equation of the circle is of the form: x2 + y2 = r2. Hence; (3)2 + (–2)2 = r2 r2 = 13 Required equation is: x2 + y2 = 13 b) Determine the equation of the tangent to the circle at the point P(3 ; – 2) The equation of the tangent through P(3 ;–2) is of the type: xx1 + yy1 = r2 x(3) + y(–2) = 13 Required equation is: 3x –2y =13 2. Determine the equation of the tangent to the circle x2 + y2 = 10 at the point K(–3 ; 1) Method 1: (Using the formula developed above) The equation of the tangent through K is of the type: xx1 + yy1 = r2 i.e: x(–3) + y(1) = 10 –3x + y = 10 y = 3x + 10 is the required equation. Method 2: First determine the gradient of the radius, OK; MOK = Mtangent = 3 (Do you recall, why?) Equation of tangent is: y -y1 = m(x – x1) y – (1) = 3(x –(–3)) y – 1 = 3x + 9 y = 3x + 10 is the required equation. Activity 2 1. Determine the equation of the tangent to the circle at the given point. Equation of circle Given point
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