JEE PYQ: Circle Question 34
Question 34 - 2019 (09 Jan Shift 1)
Three circles of radii $a, b, c$ $(a < b < c)$ touch each other externally. If they have $x$-axis as a common tangent, then:
(1) $\frac{1}{\sqrt{a}} = \frac{1}{\sqrt{b}} + \frac{1}{\sqrt{c}}$
(2) $\frac{1}{\sqrt{b}} = \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{c}}$
(3) $a, b, c$ are in A.P.
(4) $\sqrt{a}, \sqrt{b}, \sqrt{c}$ are in A.P.
Show Answer
Answer: (1)
Solution
Using Descartes’ circle theorem for circles tangent to a line: length of common external tangent between circles of radii $r_1, r_2$ touching externally $= 2\sqrt{r_1 r_2}$. The three tangent lengths sum: $2\sqrt{ab} + 2\sqrt{ac} = 2\sqrt{bc}$ (projection on x-axis). Dividing by $2\sqrt{abc}$: $\frac{1}{\sqrt{c}} + \frac{1}{\sqrt{b}} = \frac{1}{\sqrt{a}}$.
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