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Equations of Tangents to Circles

Find the equation of a tangent line to a circle and the slope of the radius from the center to the tangent line. Solve for tangents at different points on circles.

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Equations of Tangents to Circles

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  1. 1. The circle x2 + y2 = 8 has a tangent at the point (−2, 2), as shown in the diagram. Find the equation of this tangent. Find the slope of the radius from the centre of the circle to the tangent: Find the slope of the radius from the centre of the circle to the tangent: Centre: (0, 0) P:(−2, 2) (x1, y1) (x2, y2) Slope of radius = −1

  2. 1. The circle x2 + y2 = 8 has a tangent at the point (− 2, 2), as shown in the diagram. Find the equation of this tangent. Find the slope of the radius from the centre of the circle to the tangent: The radius is perpendicular to the tangent: Tangent: Slope = 1, P = (−2, 2) mradius × mtangent = −1 (y − y1) = m(x − x1) −1 × m2 = −1 y − 2 = 1(x − (−2)) m2 = 1 y − 2 = 1(x + 2) y − 2 = x + 2 0 = x − y + 4

  3. 2. The circle x2 + y2 = 20 has a tangent at the point (2, − 4). (i) Draw a sketch of the circle and tangent. Circle: x2 + y2 = 20 Centre = (0, 0)

  4. 2. The circle x2 + y2 = 20 has a tangent at the point (2, − 4). (ii) Find the equation of this tangent. Find the slope of the radius from the centre of the circle to the tangent: Centre (0, 0) P:(2, −4) (x1, y1) (x2, y2) Slope of radius = −2

  5. 2. The circle x2 + y2 = 20 has a tangent at the point (2, − 4). (ii) Find the equation of this tangent. The radius is perpendicular to the tangent: Tangent: Slope = P: (2, −4) mradius × mtangent = −1 −2 × mt = −1 m2 =

  6. 2. The circle x2 + y2 = 20 has a tangent at the point (2, − 4). (iii) Find the coordinates of the points where this tangent intersects the x and y axes. At the y-axis, x = 0: 0 = x − 2y − 10 0 = 0 − 2y − 10 2y = −10 y = −5 (0, −5) At the x-axis, y = 0: 0 = x − 2y − 10 0 = x − 2(0) − 10 0 = x − 0 − 10 10 = x (10, 0)

  7. 3. The circlex2 + y2 = 13 has a tangent at the point (2, 3). Find the equation of this tangent. Find the slope of the radius from the centre of the circle to the tangent: Centre: (0, 0)     P:(2, 3) (x1, y1)       (x2, y2)

  8. 3. The circlex2 + y2 = 13 has a tangent at the point (2, 3). Find the equation of this tangent. The radius is perpendicular to a tangent:

  9. 4. The circle (x − 5)2 + (y − 2)2 = 45 has a tangent at the point (− 1, − 1).Find the equation of this tangent. Find the slope of the radius from the centre of the circle to the tangent: Centre: (5, 2) P: (−1, −1) (x1, y1)       (x2, y2)

  10. 4. The circle (x − 5)2 + (y − 2)2 = 45 has a tangent at the point (− 1, − 1).Find the equation of this tangent. The radius is always perpendicular to the tangent: Tangent: Slope= −2, P(−1, −1) mradius × mtangent = −1 y − y1 = m(x − x1) × mt = −1 y − (−1) = −2(x − (−1)) mt = −2 y + 1 = −2(x + 1) y + 1 = −2x − 2 2x + y + 3 = 0

  11. 5. The circle (x + 1)2 + (y − 2)2 = 50 has a tangent at the point (4, − 3). Find the equation of this tangent. Find the slope of the radius from the centre of the circle to the tangent: Centre: (−1, 2) P: (4, −3) (x1, y1)        (x2, y2)

  12. 5. The circle (x + 1)2 + (y − 2)2 = 50 has a tangent at the point (4, −3). Find the equation of this tangent. The radius is perpendicular to the tangent: Tangent: Slope = 1, P(4, −3) mradius × mtangent = −1 y − y1 = m(x − x1) −1 × mt = −1 y − (−3) = 1(x − 4) mt = 1 y + 3 = 1(x − 4) y + 3 = x − 4 0 = x − y − 7

  13. 6. The circle x2 + y2 = 25 has a tangent s, at the point (−4, −3) and another tangent n, at the point (−3, 4), as shown in the diagram. (i) Find the equation of s. Find the slope of the radius from the centre of the circle to the tangent: Centre: (0, 0) P: (−4, −3) (x1, y1)       (x2, y2)

  14. 6. The circle x2 + y2 = 25 has a tangent s, at the point (−4, −3) and another tangent n, at the point (−3, 4), as shown in the diagram. (i) Find the equation of s. The radius is perpendicular to a tangent: Tangent, s: Slope = P(−4, −3) y − y1 = m(x − x1) y − (−3) = y + 3 = 3(y + 3) = −4(x + 4) 3y + 9 = −4x − 16 s: 4x + 3y + 25 = 0

  15. 6. The circle x2 + y2 = 25 has a tangent s, at the point (−4, −3) and another tangent n, at the point (−3, 4), as shown in the diagram. (ii) Find the equation of n. Find the slope of the radius from the centre of the circle to the tangent: Centre: (0, 0) P(−3, 4) (x1, y1)    (x2, y2)

  16. 6. The circle x2 + y2 = 25 has a tangent s, at the point (−4, −3) and another tangent n, at the point (−3, 4), as shown in the diagram. (ii) Find the equation of n. The radius is perpendicular to a tangent: Tangent, n: Slope = P(−3, 4) y − y1 = m(x − x1) y − 4 = (x − (−3)) y − 4 = (x + 3) 4(y − 4) = 3(x + 3) 4y − 16 = 3x + 9 0 = 3x − 4y + 25

  17. 6. The circle x2 + y2 = 25 has a tangent s, at the point (−4, −3) and another tangent n, at the point (−3, 4), as shown in the diagram. (iii) Find the equation of n. (× −3)  4x + 3y = −25 (× 4) 3x − 4y = −25 −12x − 9y = 75 Let y = 1: 12x − 16y = −100 3x − 4y = −25 − 25y = −25 3x − 4(1) = −25 y = 1 3x − 4 = −25 3x = −21 x = −7 sn = (−7, 1)

  18. 7. The circle (x − 2)2 + (y − 4)2 = 52 has a tangent k, at the point (−2, −2) and another tangent m, at the point (−2, 10). (i) Draw a diagram of this circle and the two tangents, on a coordinated plane. Centre = (2, 4)

  19. 7. The circle (x − 2)2 + (y − 4)2 = 52 has a tangent k, at the point (−2, −2) and another tangent m, at the point (−2, 10). (ii) Find the equation of k. Find the slope of the radius from the centre of the circle to the tangent: Centre: (2, 4) P (−2, −2) (x1, y1)    (x2, y2)

  20. 7. The circle (x − 2)2 + (y − 4)2 = 52 has a tangent k, at the point (−2, −2) and another tangent m, at the point (−2, 10). (ii) Find the equation of k. The radius is perpendicular to a tangent: Tangent: Slope = P(−2, −2) y − y1 = m(x − x1) y − (−2) = (x − (−2)) y + 2 = (x + 2) 3(y + 2) = −2(x + 2) 3y + 6 = −2x − 4 k: 2x + 3y + 10 = 0

  21. 7. The circle (x − 2)2 + (y − 4)2 = 52 has a tangent k, at the point (−2, −2) and another tangent m, at the point (−2, 10). (iii) Find the equation of m. Find the slope of the radius from the centre of the circle to the tangent: Centre: (2, 4) P (−2, 10) (x1, y1)    (x2, y2)

  22. 7. The circle (x − 2)2 + (y − 4)2 = 52 has a tangent k, at the point (−2, −2) and another tangent m, at the point (−2, 10). (iii) Find the equation of m. The radius is perpendicular to a tangent: Tangent: Slope = , P: (−2, 10) y − y1 = m(x − x1) y − 10 = (x − (−2)) y − 10 = (x + 2) 3(y − 10) = 2(x + 2) 3y − 30 = 2x + 4 m: 0 = 2x − 3y + 34

  23. 7. The circle (x − 2)2 + (y − 4)2 = 52 has a tangent k, at the point (− 2, − 2) and another tangent m, at the point (− 2, 10). (iv) Hence, find the point of intersection between the lines k and m. Let x = −11:  2x + 3y = −10 2x − 3y = −34 2x + 3y = −10 4x = −44 2(−11) + 3y = −10 x = −11 −22 + 3y = −10 3y = 12 y = 4 km: (−11, 4)

  24. 8. The circle chas equation x2 + y2 = 100. (i) The line t is a tangent to c at the point (− 6, 8). Find the equation of t. Find the slope of the radius from the centre of the circle to the tangent: Centre: (0, 0) P (−6, 8) (x1, y1)    (x2, y2)

  25. 8. The circle chas equation x2 + y2 = 100. (i) The line t is a tangent to c at the point (− 6, 8). Find the equation of t. The radius is perpendicular to a tangent: Tangent: P(− 6, 8) y − y1 = m(x − x1) y − 8 = (x − (− 6)) y − 8 = (x + 6) 4(y − 8) = 3 (x + 6) 4y − 32 = 3x + 18 t: 0 = 3x − 4y + 50

  26. 8. The circle chas equation x2 + y2 = 100. (ii) On a coordinated plane, draw the circle c and the tangent t. Circle: x2 + y2 = 100 Centre = (0, 0)

  27. 8. The circle chas equation x2 + y2 = 100. (iii) The line k is a tangent to c and k is parallel to the x-axis. Find the two possible equations of k. A line which is parallel to x axis has a slope = 0 At the top of the circle, y = 10 Equation of the tangent along the top of the circle: y − 10 = 0 At the bottom of the circle, y = −10 Equation of the tangent along the bottom of the circle: y + 10 = 0

  28. 9. The circlec has equation (x + 4)2 + (y − 3)2 = 29. (i) Show that the point A (−6, 8) lies on the circle c. Substitute the coordinates of (−6, 8) into the equation of the circle: (x + 4)2 + (y − 3)2 = 29 (−6 + 4)2 + (8 − 3)2 = 29 (−2)2 + 52 = 29 4 + 25 = 29 29 = 29 Since L.H.S. = R.H.S. the point A is on the circle.

  29. 9. The circlec has equation (x + 4)2 + (y − 3)2 = 29. (ii) Find the equation of k, the tangent to c at the point A. Find the slope of the radius from the centre of the circle to the tangent: Centre: (−4, 3) P (−6, 8) (x1, y1)    (x2, y2)

  30. 9. The circlec has equation (x + 4)2 + (y − 3)2 = 29. (ii) Find the equation of k, the tangent to c at the point A. Tangent: P(−6, 8) The radius is perpendicular to the tangent:     y − y1 = m(x − x1) y − 8 = (x − (−6)) y − 8 = (x + 6) 5(y − 8) = 2(x + 6) 5y − 40 = 2x + 12 k: 0 = 2x − 5y + 52

  31. 9. The circlec has equation (x + 4)2 + (y − 3)2 = 29. (iii) A second line, t, is a tangent to c at the point B such that k || t. Find the coordinates of the point B. To find the coordinates of B, go from A into the centre of the circle and out the same amount on the far side: A (−6, 8) (−4, 3) B (−2, −2)

  32. 9. The circlec has equation (x + 4)2 + (y − 3)2 = 29. (iv) Hence, or otherwise, find the equation of the line t. The tangent through B is parallel to the tangent through A m = B (−2, −2) y − y1 = m(x − x1) y − (−2) = (x − (−2)) y + 2 = (x + 2) 5(y + 2) = 2(x + 2) 5y + 10 = 2x + 4 t: 0 = 2x − 5y – 6

  33. 10. The line 2x + 3y − 13 = 0 is a tangent to a circle, with centre(−2, k) at the point (2, 3). (i) Find the slope of the tangent. 2x + 3y − 13 = 0 a = 2, b = 3

  34. 10. The line 2x + 3y − 13 = 0 is a tangent to a circle, with centre(−2, k) at the point (2, 3). (ii) Find the slope of the radius from the centre of the circle to the point of tangency. Radius  Tangent mr × mt = −1 mr × = −1 mr=

  35. 10. The line 2x + 3y − 13 = 0 is a tangent to a circle, with centre(−2, k) at the point (2, 3). (iii) Hence, find the value of k. Slope of the radius from the centre (−2, k) to the point (2, 3) is m = Centre: (−2, k) P (2, 3) (x1, y1)    (x2, y2)

  36. 10. The line 2x + 3y − 13 = 0 is a tangent to a circle, with centre(−2, k) at the point (2, 3). (iv) Hence, find the equation of the circle. Find the length of the radius, the distance from the centre to the point (2, 3). Centre: (−2, −3) P (2, 3) (x1, y1)    (x2, y2) (x − h)2 + (y − k)2 = r2 (x − (− 2))2 + (y − (− 3))2 = (x + 2)2 + (y + 3)2 = 52

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