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Example 4 - Find circle passing through (2, -2), (3, 4) - Examples

Example  4 - Chapter 11 Class 11 Conic Sections - Part 2
Example  4 - Chapter 11 Class 11 Conic Sections - Part 3 Example  4 - Chapter 11 Class 11 Conic Sections - Part 4 Example  4 - Chapter 11 Class 11 Conic Sections - Part 5


Transcript

Example 4 Find the equation of the circle which passes through the points (2, –2), and (3,4) and whose centre lies on the line x + y = 2. Equation of circle with centre (h, k) is (x – h)2 + (y – k)2 = r2 Since the circle passes through (2, – 2) Point (2, −2) will satisfy the equation of circle Putting x = 2, y = −2 in (A) (2 − h)2 + (−2 − k)2 = r2 4 + h2 − 4h + 4 + (−2)2 + (−k)2 + 2(−2)(−k) = r2 4 + h2 − 4h + 4 + k2 + 4k = r2 h2 + k2 − 4h + 4k + 8 = r2 Similarly , Since circle passes through (3, 4) Point (3, 4) will satisfy the equation of circle Putting x = 3, y = 4 in (A) (3 − h)2 + (4 − k)2 = r2 (3)2 + (h)2 − 2(3)(h) + (4)2 + (k)2 − 2(4)(k) = r2 9 + h2 − 6h + 16 + k2 − 8k = r2 h2 + k2 − 6h − 8k + 25 = r2 Subtracting (2) from (1) (h2 + k2 − 6h − 8k + 25) − (h2 + k2 − 4h + 4k + 8) = r2 − r2 h2 + k2 − 4h + 4k + 8 − h2 − k2 + 6h + 8k − 25 = 0 h2 − h2 + k2 − k2 − 4h + 6h + 4k + 8k + 8 − 25 = 0 0 + 0 + 2h + 12k − 17 = 0 2h + 12k – 17 = 0 Since center (h, k) lie on the line x + y = 2 i.e. point (h, k) will satisfy the equation of circle h + k = 2 Now our equations are 2h + 12k = 17 h + k = 2 From (4) h + k = 2 k = 2 − h Putting value of k in (3) 2h + 12(2 − h) = 17 2h + 24 − 12h = 17 −10h = 17 − 24 −10h = −7 h = (−7)/(−10) h = 0.7 Putting value of h = 0.7 in k = 2 − h k = 2 − (0.7) k = 1.3 Putting value of (h, k) = (0.7, 1.3) in (A) (x – h)2 + (y – k)2 = r2 (2 − 0.7)2 + (−2 − 1.3)2 = r2 (1.3)2 + (−3.3)2 = r2 1.69 + 10.89 = r2 12.58 = r2 r2 = 12.58 Putting value of h, k & r2 in (A) (x − 0.7)2 + (y − 1.3)2 = 12.58 Hence required equation circle is (x − 0.7)2 + (y − 1.3)2 = 12.58

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.