Circle geometry: Equations / problems

Circle Geometry Equations KUS objectives BAT Find and use the general equations of circles BAT solve geometry problems using your knowledge of circles Starter: (next page)

Substitute y = 3x – 10 into equation of circle WB 16 Show that the line y = 3x – 10 is a tangent to the circle 𝑥 2 + 𝑦 2 =10 Substitute y = 3x – 10 into equation of circle 𝑥 2 + (3𝑥−10) 2 =10 10𝑥 2 −60𝑥+90=0 𝑥 2 −6𝑥+9=0 (𝑥−3) 2 =0 Since this has a repeated root there is only one point where the line meets The circle. So the line is a tangent to the circle

The equation of a circle is 𝑥 2 + 𝑦 2 −4𝑥+2𝑦=15 WB 17 The equation of a circle is 𝑥 2 + 𝑦 2 −4𝑥+2𝑦=15 a) Find the coordinates of the centre C of the circle, and the radius of the circle b) Show that the point P (4, -5) lies on the circle. c) Find the equation of the tangent to the circle at the point P. a) Rearrange to (𝑥−2) 2 + (𝑦+1) 2 −4−1=15 (𝑥−2) 2 + (𝑦+1) 2 =20 Center 2, −1 radius 20 =2 5 b) Substituting values of point P gives (4) 2 + (−5) 2 −4 4 +2 −5 =16+26 −16 −10=15 QED c) Gradient of CP is −1−−5 2−4 =−2 so gradient of tangent is 1 2 Equation of tangent through P is 𝑦−−5= 1 2 (𝑥−4) Or 𝑥−2𝑦=14

AB is the diameter of a circle. A is 1, 3 B is 7, −1 WB 18a AB is the diameter of a circle. A is 1, 3 B is 7, −1 Find the coordinates of the centre C of the circle Find the radius of the circle Find the equation of the circle. The line y + 5x = 8 cuts the circle at A and again at a second point D. Calculate the coordinates of D. [4] e) Prove that the line AB is perpendicular to the line CD. a) C is the midpoint AB to C (4, 1) b) Using Pythagoras distance CA (4−1) 2 + (1−3) 2 = 13 c) Using centre and radius (𝑥−4) 2 + (𝑦−1) 2 = 13 d) Substituting 𝑦=8−5𝑥 into equation of circle gives (𝑥−4) 2 + (8−5𝑥−1) 2 = 13 → 𝑥 2 −8𝑥+16 +( 25 2 −70𝑥+49)= 13 → 26𝑥 2 −78𝑥+52= 0 → 𝑥 2 −3𝑥+2= 0

AB is the diameter of a circle. A is 1, 3 B is 7, −1 WB 18b AB is the diameter of a circle. A is 1, 3 B is 7, −1 Find the coordinates of the centre C of the circle Find the radius of the circle Find the equation of the circle. The line y + 5x = 8 cuts the circle at A and again at a second point D. Calculate the coordinates of D. [4] e) Prove that the line AB is perpendicular to the line CD. d) Substituting 𝑦=8−5𝑥 into equation of circle gives → 𝑥 2 −3𝑥+2= 0 → (𝑥−1)(𝑥−2)= 0 x-value at point D is x= 2 and so y=8−5 2 =−2 point D is (2, −2) e) Gradient AB is 3−−1 1−7 =− 2 3 Gradient CD is 1−−2 4−2 =+ 3 2 product of gradients is − 2 3 × 3 2 =−1 so AB and CD are perpendicular

One thing to improve is – KUS objectives BAT Find and use equations of circles BAT find the centre and radius of a circle from its equation self-assess One thing learned is – One thing to improve is –

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