Let two circles having radii r1 and r2 are orthogonal to each other. If the length of their common chord is k times the square root of harmonic mean between the squares of their radii, then k4 is equal to

Question

Let two circles having radii r1 and r2 are orthogonal to each other. If the length of their common chord is k times the square root of harmonic mean between the squares of their radii, then k4 is equal to (A) 13 (B) 7 (C) 4 (D) 2

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/m-3icarnvokhct.png" /> Area of Δ ABC =(1/2)r1r2=(1/2)((l/2))√r12 + r22 ⇒ l=(2 r1 r2/√r12 + r22) ⇒ l=√2√(2 r12 r22/r12 + r22)⇒ k=√2 ⇒ k4=4

Q. Let two circles having radii $r_{1}$ and $r_{2}$ are orthogonal to each other. If the length of their common chord is $k$ times the square root of harmonic mean between the squares of their radii, then $k^{4}$ is equal to

$13$

$7$

$4$

$2$

Area of $Δ A BC$
$= \frac{1}{2} r_{1} r_{2} = \frac{1}{2} (\frac{l}{2}) r_{1}^{2} + r_{2}^{2}$
$\Rightarrow l = \frac{2 r _{1} r _{2}}{r _{1}^{2} + r _{2}^{2}}$
$\Rightarrow l = 2 \frac{2 r _{1}^{2} r _{2}^{2}}{r _{1}^{2} + r _{2}^{2}} \Rightarrow k = 2$
$\Rightarrow k^{4} = 4$

Q. Let two circles having radii r1​ and r2​ are orthogonal to each other. If the length of their common chord is k times the square root of harmonic mean between the squares of their radii, then k4 is equal to

13

7

4

2

Area of ΔABC =21​r1​r2​=21​(2l​)r12​+r22​​ ⇒l=r12​+r22​​2r1​r2​​ ⇒l=2​r12​+r22​2r12​r22​​​⇒k=2​ ⇒k4=4

Q. Let two circles having radii $r_{1}$ and $r_{2}$ are orthogonal to each other. If the length of their common chord is $k$ times the square root of harmonic mean between the squares of their radii, then $k^{4}$ is equal to

$13$

$7$

$4$

$2$

Area of $Δ A BC$
$= \frac{1}{2} r_{1} r_{2} = \frac{1}{2} (\frac{l}{2}) r_{1}^{2} + r_{2}^{2}$
$\Rightarrow l = \frac{2 r _{1} r _{2}}{r _{1}^{2} + r _{2}^{2}}$
$\Rightarrow l = 2 \frac{2 r _{1}^{2} r _{2}^{2}}{r _{1}^{2} + r _{2}^{2}} \Rightarrow k = 2$
$\Rightarrow k^{4} = 4$