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(B) The centers of the circles are $O_1(0, 0)$ and $O_2(h, 0)$,and both have radius $r = 1$.Let the transverse common tangent touch the circles at $A$

Vedclass · Accepted Answer

$\pm 4$

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(B) The centers of the circles are $O_1(0, 0)$ and $O_2(h, 0)$,and both have radius $r = 1$.Let the transverse common tangent touch the circles at $A$ and $B$,intersecting the line joining the centers at $P$.Since the radii are equal,$P$ is the midpoint of $O_1O_2$ and also the midpoint of $AB$.Given the length of the transverse common tangent $AB = 2\sqrt{3}$,we have $AP = PB = \sqrt{3}$.In the right-angled triangle $\Delta O_1AP$,we have $O_1A = r = 1$ and $AP = \sqrt{3}$.By the Pythagorean theorem,$O_1P^2 = O_1A^2 + AP^2 = 1^2 + (\sqrt{3})^2 = 1 + 3 = 4$.Thus,$O_1P = 2$.Since $P$ is the midpoint of $O_1O_2$,the distance $O_1O_2 = 2 	imes O_1P = 2 	imes 2 = 4$.Since $O_1 = (0, 0)$ and $O_2 = (h, 0)$,the distance $O_1O_2 = |h| = 4$,which implies $h = \pm 4$.

If the length of the transverse common tangent to the circles $x^{2} + y^{2} = 1$ and $(x - h)^{2} + y^{2} = 1$ is $2\sqrt{3}$,find the value of $h$.

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