O is the centre of the circle, with OM AB. OP is the O is the centre of the circle, O is the centre of the circle, radius of the circle. with OM AB. OP is the with OM AB. OP is the radius of the circle. radius of the circle. In the above figure is a In the above In the above figure is a AB AB does AB secant to the circle. not intersect the circle. tangent to the circle. Note that AB: Intersects the circle in two places (see if OM < r Figure 1) Does not intersect the circle (see if OM > Figure 2) r Intersects the circle in one and only one place (see if OM = r Figure 3) Worked examples 1. Show that 2x + + 5 = 0 is a tangent to the circle defined by: + = 5 y x2 y2 Proving that there are two equal roots the given straight-line equation in the form y = + c, mx i.e: = 2x 5 y – – Substitute = 5 in + = 5 y x2 y2 –2x – i.e: + (–2x = 5 x2 –5)2 4x2 20x 25 5 x2 + + + = 0 – 5x2 20x 20 0 + + = 5(x2 4x 4) 0 + + = 2)2 0 (x + = = 2 is a repeated root, which implies that the straight line: 2x + + 5 = 0 is a x y – tangent to the given circle. Can you solve the given problem differently? Discuss with your partner. 3.2 The equation of the tangent through a point (x1 ; on y1) the circle + = x2 y2 r2
O is the centre of the circle, with OM AB. OP is the radius of the circle. O is the centre of the circle, with OM AB. OP is the radius of the circle. O is the centre of the circle, with OM AB. OP is the radius of the circle. In the above figure AB is a secant to the circle. In the above AB does not intersect the circle. In the above figure AB is a tangent to the circle. Note that AB: Intersects the circle in two places (see Figure 1) if OM < r Does not intersect the circle (see Figure 2) if OM > r Intersects the circle in one and only one place (see Figure 3) if OM = r Worked examples 1. Show that 2x + y + 5 = 0 is a tangent to the circle defined by: x2 + y2 = 5 Proving that there are two equal roots the given straight-line equation in the form y = mx + c, i.e: y = – 2x – 5 Substitute y = –2x – 5 in x2 + y2 = 5 i.e: x2 + (–2x –5)2 = 5 x2 + 4x2 + 20x + 25 – 5 = 0 5x2 + 20x + 20 = 0 5(x2 + 4x + 4) = 0 (x + 2)2 = 0 x = – 2 is a repeated root, which implies that the straight line: 2x + y + 5 = 0 is a tangent to the given circle. Can you solve the given problem differently? Discuss with your partner. 3.2 The equation of the tangent through a point (x1 ; y1) on the circle x2 + y2 = r2
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