Question

If r1 and r2 are the radii of two concentric circles (r1 is greater than r2 ). If the chord of the larger circle ab touches the inner circke find AB​

Answer 1

Given: r1 and r2 are the radii of two concentric circles and r1> r2  

AB is chord of larger circle touches the inner circle

 

To Find: Length of AB

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• AB is chord of circle  and touches other circle,

        let say at M  and O is centre of both circles.

Then OM ⊥ AB (as AB is tangent)  

Also AB is chord  hence M is mid point of AB  

$\sf AM^2 = OA^2 - OM^2$

 

$\sf OA = r_1 ( radius ~of ~larger ~circle) \\\\\sf OM = r_2 (\sf radius \;of \; smaller \; circle) \\\\\implies \sf AM^2 = r_1^2 - r_2^2 \\\\\sf AM = \sf\sqrt{( \sf r_1^2 - r_2^2)} \\\\ \sf AB = 2AM = 2 \sqrt{(\sf r_1^2 - r_2^2)}$

$\therefore \underline{\pink{\sf AB = \sqrt{r_1^2 - r_2^2}}}$

Answer 2

$\huge \mid \star \:\cal \colorbox{aqua}{ANSWER} \: \star \mid$

AB is chord of circle and touches outer circle.

Let us assume that M and O is center of both the circle.

Than OM _ AB

Also AB is chord hence M is mid point of AB.

AM² = OA² - OM²

OA = r1 ( radius of larger circle)

OM = r2 (radius of smaller circle)

AM ² = r1² - r2²

AM = (√r1²-r2²)

AB = 2AM = 2(√r1²-r2²)

Answer »» AB = (√r1²-r2²)