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Length of Chord of Contact

Assertion :Th...

Question

Assertion :The number of common tangents to the circle $x^{2} + y^{2} = 4$ & $x^{2} + y^{2} - 8 x - 6 y - 24 = 0$ is $4$ . Reason: Circles with centre $c_{1}, c_{2}$ & radii $r_{1}, r_{2}$ and if $| c_{1} c_{2} | > r_{1} + r_{2}$ , then circles have $4$ common tangents.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is D Assertion is incorrect but Reason is correct
$x^{2} + y^{2} = 4$ . $∴ c_{1} = (0, 0) . r_{1} = 2$
and $x^{2} + y^{2} - 8 x - 6 y - 24 = 0$
$∴ c_{2} = (4, 3), r_{2} = 7$
$∴ c_{1} c_{2} = 5$ & $| r_{2} - r_{1} | = 5$
Now $c_{1} c_{2} = | r_{2} - r_{1} | \Rightarrow$ Circles touch internally.
$∴$ only one tangent is possible.
Hence Assertion (A) is false but Reason (R) is true

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