Engineering
Mathematics
Family of Circle
Question

Circle(s) touching x‑axis at a distance 3 from the origin and having an intercept of length 27  on y‑axis is(are)

x2 + y2 – 6x + 8y + 9 = 0

x2 + y2 – 6x + 7y + 9 = 0

x2 + y2 – 6x – 8y + 9 = 0

x2 + y2 – 6x – 7y + 9 = 0

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Solution

Let the equation of circle be

 

Put   x = 0

            ⇒         y2 + λy + 9 = 0

                     | y1 – y2 | = 27

            ⇒         (λ)2 – 36 = 28    ⇒        λ2 = 64             ⇒         λ = ± 8

            So, equation of circles is x2 + y2 – 6x ± 8y + 9 = 0

            Similarly, equation of other circle will be

                        x2 + y2 + 6x ± 8y + 9 = 0.