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Position of a Point W.R.T Hyperbola

The number of...

Question

The number of common tangents to the circles $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} - 6 x - 8 y + 24 = 0$ is :

A

3

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B

4

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C

2

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D

1

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Solution

The correct option is A $4$
Equation of circles are $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} - 6 x - 8 y + 24 = 0$

The centres and radii of circles are $C_{1} (0, 0), C_{2} (3, 4)$ and
$r_{1} \equiv 2, r_{2} \equiv \sqrt{9 + 16 - 24} = 1$
Now, $C_{1} C_{2} = \sqrt{(3 - 0)^{2} + (4 - 0)^{2}} = 5$
And, $r_{1} + r_{2} = 2 + 1 = 3$
Since, $C_{1} C_{2} > r_{1} + r_{2}$

$∴$ The circles do not touch each other
Hence, the number of common tangents $= 4$ .

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