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Chord Properties of Circles

Assertion :Ci...

Question

Assertion :Circles $x^{2} + y^{2} = 4$ & $x^{2} + y^{2} - 8 x + 7 = 0$ intersect each other in two distinct points. Reason: Circles with centres $c_{1}$ & $c_{2}$ and radius $r_{1}$ & $r_{2}$ intersect at two distinct points if $| r_{1} - r_{2} | < c_{1} c_{2} < r_{1} + r_{2}$

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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Both Assertion and Reason are incorrect

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
$x^{2} + y^{2} = 4$ $∴ c_{1} = (0, 0), r_{1} = 2$
$x^{2} + y^{2} - 8 x + 7 = 0$ $∴ c_{2} = (4, 0), r_{2} = 3$
$∴ | c_{1} c_{2} | = 4$ & $| r_{1} - r_{2} | = 1, r_{1} + r_{2} = 5$
$| r_{1} - r_{2} | < | c_{1} c_{2} | < r_{1} + r_{2}$ i.e circles intersect
$\Rightarrow$ Assertion (A) & Reason (R) are true & Reason (R) is correct explanation of Assertion (A)
Ans: A

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Similar questions

x^{2} + y^{2} = 4

x^{2} + y^{2} - 8 x + 7 = 0

C_{1}

C_{2}

r_{1}

r_{2}

| C_{1} C_{2} | < r_{1} + r_{2}

x^{2} + y^{2} = 4

x^{2} + y^{2} - 8 x - 6 y - 24 = 0

4

c_{1}, c_{2}

r_{1}, r_{2}

| c_{1} c_{2} | > r_{1} + r_{2}

4

x^{2} + y^{2} + 2 a x + c = 0

x^{2} + y^{2} + 2 b y + c = 0

\frac{1}{c} = \frac{1}{a^{2}} + \frac{1}{b^{2}}

c_{1}, c_{2}

r_{1}, r_{2}

c_{1} + c_{2} = r_{1} \pm r_{2}

Assertion: All the points (1, 0), (-1, 0), (2, 0) and (5, 0) lie on the x – axis.

Reason: Equation of the x – axis is y = 0
Now, choose the correct option.

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