The locus of centre of circle touching the circles (x + 3)2 + y2 = 16 and (x

Engineering

Mathematics

Locus of a Point

Family of Circle

Standard Hyperbola

Question

The locus of centre of circle touching the circles (x + 3)² + y² = 16 and (x – 3)² + y² = 4 externally is

$\frac{x^{2}}{8} - y^{2} = 1$

$x^{2} - \frac{y^{2}}{8} = 1$

$x^{2} - \frac{y^{2}}{4} = 1$

$\frac{x^{2}}{4} - y^{2} = 1$

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Let the equation of variable circle be
       (x – h)² + (y – k)² = r²        ……(1)
Now, according to question
       CC₁ = r + 4
       CC₂ = r + 2
$∴$ |CC₁ – CC₂| = 2
⇒ The locus of P(h, k) s a hyperbola with foci C₁ (–3, 0) & C₂ (3, 0) and transverse axis of length 2.
Also, $2 a \cdot e_{h} = 6 \Rightarrow e_{h} = \frac{6}{2} = 3$
Also,   $e_{h}^{2} = 1 + \frac{b^{2}}{a^{2}} \Rightarrow 9 - 1 = \frac{b^{2}}{1} \Rightarrow b^{2} = 8$
So, equation of hyperbola is $\frac{x^{2}}{1} - \frac{y^{2}}{8} = 1$