Ex 10.1, 10 - Chapter 10 Class 11 Conic Sections
Last updated at April 17, 2024 by Teachoo
Ex 10.1
Last updated at April 17, 2024 by Teachoo
Ex 10.1, 10 Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x + y = 16. Let the equation of the circle be (x – h)2 + (y – k)2 = r2. Since circle passes through (4, 1) It will satisfy the equation of circle Putting x = 4 & y = 1 in (A) (x – h)2 + (y – k)2 = r2. (4 − h)2 + (1 − k)2 = r2 42 + h2 − 8h + 12 + k2 − 2k = r2 16 + h2 − 8h + 1 + k2 − 2k = r2 h2 + k2 − 8h − 2k + 17 = r2 Since circle passes through (6, 5) It will satisfy the equation of circle Putting x = 6, & y = 5 in (A) (x – h)2 + (y – k)2 = r2 (6 – h)2 + (5 – k)2 = r2 62 + h2 − 12h + 52 + k2 − 10k = r2 36 + h2 − 12h + 25 + k2 − 10k = r2 h2 + k2 − 12h − 10k + 36 + 25 = r2 h2 + k2 − 12h − 10k + 61 = r2 Since centre of circle (h, k) lie on the line 4x + y = 16 i.e. centre (h, k) will satisfy the equation of line 4x + y = 16 Putting (h, k) in 4x + y = 16 4h + k = 16 Solving (1) & (2) h2 + k2 − 12h − 10k + 61 = r2 …(1) h2 + k2 − 8h − 2k + 17 = r2 …(2) Subtracting (2) from (1) (h2 + k2 − 8h − 2k + 17) − (h2 + k2 − 12h − 10k + 61)= r2 −r2 h2 + k2 − 8h − 2k + 17 − h2 − k2 + 12h + 10k − 61 = 0 h2 − h2 + k2 − k2 − 8h + 12h − 2k + 10k + 17 − 61 = 0 0 + 0 + 4h + 8k − 44 = 0 4h + 8k − 44 = 0 4h + 8k = 44 h + 2k = 11 Now our equations are 4h + k = 16 …(3) h + 2k = 11 …(4) From (4) h + 2k = 11 h = 11 – 2k Putting value of h in (3) 4(11 − 2k) + k = 16 44 − 8k + k = 16 44 − 7k = 16 −7k = 16 − 44 −7k = − 28 k = (−28)/(−7) k = 4 Putting value of k = 4 in (4) h + 2k = 11 h = 11 − 2(4) h = 11 − 8 h = 3 Hence, h = 3 & k = 4 Putting value of h, k in (1) h2 + k2 − 8h − 2k + 17 = r2 32 + 42 − 8(3) − 2(4) + 17 = r2 9 + 16 − 24 − 8 + 17 = r2 10 = r2 r2 = 10 Equation of a circle is (x − h) 2 + (y − k) 2 = r2 Putting values (x − 3) 2 + (y − 4) 2 = 10 x2 + 32 − 2(x)(3) + y2 + 42 − 2(y)(4) = 10 x2 + 9 − 6x + y2 + 16 − 8y = 10 x2 + y2 − 6x − 8y + 9 + 16 − 10 = 0 x2 + y2 − 6x − 8y + 15 = 0 Which is the required equation of circle