Consider a circle S with centre at the origin and radius 4. Four circles A, B, C and D each with radius unity and centres (–3, 0), (–1, 0), (1, 0) and (3, 0) respectively are drawn. A chord PQ of the circle S touches the circle B and passes through the centre of the circle C. If the length of this chord can be expressed as \sqrt{x} then find x

Engineering

Mathematics

Common Tangent To two Circle

Question

Consider a circle S with centre at the origin and radius 4. Four circles A, B, C and D each with radius unity and centres (–3, 0), (–1, 0), (1, 0) and (3, 0) respectively are drawn. A chord PQ of the circle S touches the circle B and passes through the centre of the circle C. If the length of this chord can be expressed as $\sqrt{x}$ then find x

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Note that triangles BCM and OCN are similar
now let ON = p. N will be mid point of chord PQ
$∴$ $\frac{p}{1} = \frac{1}{2} \Rightarrow p = \frac{1}{2}$
now $2 L = 2 \sqrt{r^{2} - p^{2}}$ for large circle (r = 4)
$= 2 \sqrt{16 - (1 / 4)} = \sqrt{63}$

Alternatively:   Equation of large circle as x² + y² = 16
now   C = (1, 0) with slope $PQ = - \frac{1}{\sqrt{3}}$ (think !)
equation of PQ : $\sqrt{3}$ y + x = 1
P (from origin) = $\frac{1}{2}$    ⇒   result

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