Question

there is a common chord of 2 circles with radius 15 and 20 the distance between the two centers is 25 the length of the chord is 6xk , find k​

answer of value of k=3
but I want to know detailed solution

Answer 1

Answer:

answer of k=4

Let the length of half of the chord be 'y'

√(15^2-y^2)+√(20^2-y^2)=25

Squaring both sides,

225-y^2+400-y^2+2√(225-y^2)(400-y^2)=625

2√(225-y^2)(400-y^2)=2y^2

Squaring both sides,

(225-y^2)(400-y^2)=y^4

225×400-225y^2-400y^2+y^4=y^4

225×400=625×y^2

y^2=(225×400)/625

y=(15×20)/25

y=12

Length of the chord= 2y=2×12=24

According to the question,

6× k=24

k=4

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Answer 2

Answer:

k = 4 cm

Step-by-step explanation:

Note :

The radii of both the circles and the line joining the centres of the two circles form a right angled triangle. So, length of common chord is twice the length of its altitude dropped from vertex to the hypotenuse.

Now,

Given, Distance between the two centers is 25 cm.

Let the altitude be h and hypotenuse be x and 25 - x cm.

So,

h² + x² = 15² and h² + (25 - x)² = 20²

⇒ 225 - x²  = 400 - x² + 50x - 625

⇒ 50x = 450

⇒ x = 9

Then,

h² + x² = 15²

⇒ h² + 9² = 225

⇒ h² = 144

⇒ h = 12 cm

Thus, length of chord = 24 cm

But,

Given, Length of chord = 24

⇒ 6xk = 24[∵ h = x]

⇒ k = 4 cm

Therefore, k = 4 cm

Hope it helps!