Consider the two circles x2+y2=9 and x2+y2−10x+9=0Statement-1:If the length of the common chord of the circles is λ5 , λ equals 24.Statement-2: If θ is the angle between the common tangents, then cosθ=ab (a,b are co-prime) , where b−a equals 2.

Statement 1: The circles are orthogonal.Length of common chord is 2.125=245 Statement 2:Since, the tangents meet at (-15,0).∴sinθ2=315=15 and cosθ=1−2sin2θ2=2325

Consider the two circles x2+y2=9 and x2+y2−10x+9=0Statement-1:If the length of the common chord of the circles is λ5 , λ equals 24.Statement-2: If θ is the angle between the common tangents, then cosθ=ab (a,b are co-prime) , where b−a equals 2.

Statement 1: The circles are orthogonal.Length of common chord is 2.125=245 Statement 2:Since, the tangents meet at (-15,0).∴sinθ2=315=15 and cosθ=1−2sin2θ2=2325

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Consider the two circles $x^{2} + y^{2} = 9$ and $x^{2} + y^{2} - 10 x + 9 = 0$
Statement-1:If the length of the common chord of the circles is $\frac{λ}{5}$ , $λ$ equals 24.
Statement-2: If $θ$ is the angle between the common tangents, then $\cos θ = \frac{a}{b}$ (a,b are co-prime) , where $b - a$ equals 2.

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Statement 1 is false, Statement 2 is false

Statement 1 is true, Statement 2 is true

Statement 1 is true, Statement 2 is false

Statement 1 is false, Statement 2 is true

answer is D.

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Detailed Solution

Statement 1: The circles are orthogonal.
Length of common chord is $2. \frac{12}{5} = \frac{24}{5}$
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Statement 2:Since, the tangents meet at (-15,0).

$∴ \sin \frac{θ}{2} = \frac{3}{15} = \frac{1}{5}$ and $\cos θ = 1 - 2 \sin^{2} \frac{θ}{2} = \frac{23}{25}$