In the xy-plane above, the circle has a center (h, k)and radius 10. What is the value of k?

The correct answer is: 6

○        The concept used in the question is the concept of the circle.
○        The general equation of a circle is given by  (x − h)² + (y − k)² = r², where (h,k) is the center of the circle and r is the radius of the circle.

• Step by step explanation:

○        Given: 
Circle has centre at (h,k)
Radius = 10
Circle passes through (4,0) and (20,0)
○        Step 1:
equation of a circle.
 (x − h)² + (y − k)² = r²
(x − h)² + (y − k)² = 100    ( as r = 10 )
○        Step 2:
○        The Circle passes through (4,0) and (20,0).
Therefore,
(4,0) and (20,0) satisfy equation (x − h)² + (y − k)² = 100
∴ (4 − h)² + (0 − k)² = 100
4² - 2(4)h + h² + k² = 100
16  - 8h + h² + k² = 100
h² + k² = 100 - 16 + 8h
h² + k² = 84 + 8h —--------- equation 1.

(x − h)² + (y − k)² = 100
∴ (20 − h)² + (0 − k)² = 100
20² - 2(20)h + h² + k² = 100
400  - 40h + h² + k² = 100
h² + k² = 100 - 400 + 40h
h² + k² =  40h - 300 —--------- equation 2.

○        Step 3:
○        Equate equation 1 and 2.
○        We get,
84 + 8h =  40h - 300
84 + 300 = 40h - 8h
384 = 32h
= h
12 = h
○        Step 4:
○        Put h = 12 in equation 1.
h² + k² = 84 + 8h
12² + k² = 84 + 8(12)
144+ k² = 84 + 96
k² = 180 - 144
k² = 36
k = 6
As centre is in 1st quadrant therefore k = 6.

• Final Answer:

Hence, value of k is 6.