JEE Main · 2020 · 09 Jan 2020 · Shift 2 · Circle

If the curves,

x^{2} - 6 x + y^{2} + 8 = 0

and

x^{2} - 8 y + y^{2} + 16 - k = 0, (k > 0)

touch each other at a point, then the largest value of

k

is ______.

Answer: 36

Solution

Completing the squares, the first circle is

(x - 3)^{2} + y^{2} = 1

, centre

(3, 0)

, radius

1

, and the second is

x^{2} + (y - 4)^{2} = k

, centre

(0, 4)

, radius

k

. The centres are

3^{2} + 4^{2} = 5

apart. The circles touch externally when

1 + k = 5

, giving

k = 16

, and internally when

∣ k - 1∣ = 5

, giving

k = 6

and

k = 36

. The largest value is

k = 36

JEE Main · 2020 · 09 Jan 2020 · Shift 2 · Integer answer

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