Ex 11.1, 13 - Circle passing through (0, 0), making intercepts a - Circle

  1. Chapter 11 Class 11 Conic Sections
  2. Serial order wise
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Ex 11.1, 13 Find the equation of the circle passing through (0, 0) and making intercepts a and b on the coordinate axes. Let the equation of circle be (x – h)2 + (y – k)2 = r2 where (h, k) is the centre & r is the radius of a circle Also given that circle making intercepts a & b on the coordinate axes Let intercept on x-axis be a, So, coordinates of point A (a, 0) Let intercept on y-axis be b, So, coordinates of point B (0, b) Now, given that circle passes through (0, 0) So, Point (0, 0) will satisfy the equation of circle Putting x = 0 & y = 0 in (1) (0 − h)2 + (0 − k)2 = r2 h2 + k2 = r2 Also, point A (a, 0) lies on circle So, point A(a, 0) will satisfy the equation of circle Putting x = a & y = 0 in (1) (a − h)2 + (0 − k)2 = r2 (a − h)2 + k2 = r2 a2 + h2 − 2ah + k2 = r2 h2 + k2 + a2 − 2ah = r2 Putting h2 + k2 = r2 from (2) r2 + a2 − 2ah = r2 a2 − 2ah = r2 − r2 a(a − 2h) = 0 a − 2h = 0 h = ﷐𝑎 ﷮2﷯ Similarly, Point B(0, b) lie on the circle Point B(0, b) will satisfy the equation of circle Putting x = 0 & y = b in (1) (0 – h)2 + (b – k)2 = r2 h2 + (b – k)2 = r2 h2 + b2 + k2 − 2bk = r2 h2 + k2 + b2 − 2bk = r2 Putting h2 + k2 = r2 from (2) r2 + b2 − 2ah = r2 b2 − 2bk = r2 − r2 b(b − 2k) = 0 b − 2k = 0 b = 2k 2k = b k = ﷐𝑏 ﷮2﷯ Thus, h = ﷐𝑎 ﷮2﷯ & k = ﷐𝑏 ﷮2﷯ So, C(h, k) = ﷐﷐𝑎﷮2﷯, ﷐𝑏﷮2﷯﷯ Now we need to find radius of circle Putting h = ﷐𝑎﷮2﷯ & b = ﷐𝑏﷮2﷯ in (2) h2 + k2 = r2 ﷐﷐﷐𝑎﷮2﷯﷯﷮2﷯ + ﷐﷐﷐𝑏﷮2﷯﷯﷮2﷯ = r2 ﷐﷐𝑎﷮2﷯﷮4﷯ + ﷐﷐𝑏﷮2﷯﷮4﷯ = r2 ﷐﷐﷐𝑎﷮2﷯ + 𝑏﷮2﷯﷮4﷯ = r2 r2 = ﷐﷐﷐𝑎﷮2﷯ + 𝑏﷮2﷯﷮4﷯ Putting value of (h, k) & r2 in (1) (x – h)2 + (y – k)2 = r2 ﷐﷐𝑥− ﷐𝑎﷮2﷯﷯﷮2﷯ + ﷐﷐𝑦−﷐𝑏﷮2﷯﷯﷮2﷯ = ﷐﷐﷐𝑎﷮2﷯ + 𝑏﷮2﷯﷮4﷯ (x)2 + ﷐﷐﷐𝑎﷮2﷯﷯﷮2﷯ − 2(x) ﷐﷐𝑎﷮2﷯﷯ + y2 +﷐﷐﷐𝑏﷮2﷯﷯﷮2﷯ − 2(y) ﷐﷐𝑏﷮2﷯﷯ = ﷐﷐﷐𝑎﷮2﷯ + 𝑏﷮2﷯﷮4﷯ x2 + ﷐﷐𝑎﷮2﷯﷮4﷯ − ax + y2 + ﷐﷐𝑏﷮2﷯﷮4﷯ − by = ﷐﷐﷐𝑎﷮2﷯ + 𝑏﷮2﷯﷮4﷯ x2 + y2 − ax − by + ﷐﷐𝑎﷮2﷯﷮4﷯ + ﷐﷐𝑏﷮2﷯﷮4﷯ = ﷐﷐﷐𝑎﷮2﷯ + 𝑏﷮2﷯﷮4﷯ x2 + y2 − ax − by + ﷐﷐﷐𝑎﷮2﷯ + 𝑏﷮2﷯﷮4﷯ = ﷐﷐﷐𝑎﷮2﷯ + 𝑏﷮2﷯﷮4﷯ x2 + y2 − ax − by = ﷐𝑎2 + 𝑏2﷮4﷯ − ﷐﷐﷐𝑎﷮2﷯ + 𝑏﷮2﷯﷮4﷯ x2 + y2 − ax − by = 0 Which is required equation of circle

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