Assertion :Circles $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} - 8 x + 7 = 0$ intersect each other at two distinct points because Reason: Circles with centres $C_{1}$ and $C_{2}$ and radii $r_{1}$ and $r_{2}$ intersect at two distance points, if $| C_{1} C_{2} | < r_{1} + r_{2}$

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 C Assertion is correct but Reason is incorrect
circles intersect at two distinct points when $| r_{1} - r_{2} | < C_{1} C_{2} < | r_{1} + r_{2} |$ ...(1)
now, $C_{1} (0, 0)$ and $r_{1} = 2$
and $C_{1} (4, 0)$ and $r_{1} = 3$
$C_{1} C_{2} = 4$ , $| r_{1} - r_{2} | = 1$ and $| r_{1} + r_{2} | = 5$
condition 1 is satisfied
Therefore, they intersect in two distinct points
Ans: C

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Similar questions

Q. 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}

Q. Assertion: If two circles intersect at two points, then the line joining their centres is perpendicular to the common chord.
Reason: The perpendicular bisectors of two chords of a circle intersect at its centre.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

Q. Assertion: The compressibility factor for

H_{2}

and

H e

(1 + \frac{P b}{R T})

Reason: The compressibility factor for

H_{2}

and

H e

can be derived from Van der Waal's equation.

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.

Assertion (A) Boiling points of alcohols and ethers are high.
Reason (R) They can form intermolecular hydrogen-bonding.
(a) Assertion and reason both are correct and reason is correct explanation of assertion.
(b) Assertion and reason both are wrong statements.
(c) Assertion is correct statement but reason is wrong statement.
(d) Assertion is wrong statement but reason is correct statement.
(e) Both assertion and reason are correct statements but reason is not correct explanation of assertion.

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Assertion :Circles x2+y2=4 and x2+y2−8x+7=0 intersect each other at two distinct points because Reason: Circles with centres C1 and C2 and radii r1 and r2 intersect at two distance points, if |C1C2|<r1+r2

The correct option is C Assertion is correct but Reason is incorrectcircles intersect at two distinct points when |r1−r2|<C1C2<|r1+r2| ...(1)now, C1(0,0) and r1=2and C1(4,0) and r1=3C1C2=4, |r1−r2|=1 and |r1+r2|=5condition 1 is satisfiedTherefore, they intersect in two distinct pointsAns: C

Assertion :Circles $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} - 8 x + 7 = 0$ intersect each other at two distinct points because Reason: Circles with centres $C_{1}$ and $C_{2}$ and radii $r_{1}$ and $r_{2}$ intersect at two distance points, if $| C_{1} C_{2} | < r_{1} + r_{2}$