First slide

Hyperbola in conic sections

Question

$If two circles (x + 4)^{2} + y^{2} = 1 and (x - 4)^{2} + y^{2} = 9 are touched externally by a variable circle, then locus of centre of variable circle is$

Moderate

A

$\frac{x^{2}}{15} - \frac{y^{2}}{1} = 1$
B

$\frac{x^{2}}{4} - \frac{y^{2}}{12} = 1$
C

$\frac{x^{2}}{12} - \frac{y^{2}}{4} = 1$
D

$\frac{x^{2}}{1} - \frac{y^{2}}{15} = 1$

Solution

$\begin{array}{l} CS = r + 1 \\ {CS}^{'} = r + 3 \\ CS - {CS}^{'} = 2 \end{array}$
Locus of C is hyperbola with (-4,0) and (4, 0) as foci. Here $2 a = 2, so a = 1$
$and 2 a e = 8, so a e = 4$
$b^{2} = a^{2} (e^{2} - 1) = 15$

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