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UPSC NDA I 2026 Mathematics Paper

For the next two (02) items that follow:

Consider the points

\mathrm{A}(0,2), \mathrm{B}(2,3)

C(4,5)

and

D(0, k)

Question : 59 of 120

Marks: +1, -0

If the points lie on a circle, then what is/are the possible value(s) of k ?

Solution:

Concept:

Find the circle passing through

A, B, C

; then check which values of

k

make

D

lie on it.

Explanation:

Let the circle be

x^2 + y^2 + ux + vy + w = 0

Using

A(0,2)

4 + 2v + w = 0

Using

B(2,3)

13 + 2u + 3v + w = 0

Using

C(4,5)

41 + 4u + 5v + w = 0

Solving:

u = 5

v = -19

w = 34

So the circle is

x^2 + y^2 + 5x - 19y + 34 = 0

Since

D(0,k)

lies on it,

k^2 - 19k + 34 = 0

Factorising:

(k - 2)(k - 17) = 0

Hence

k = 2

k = 17

Answer:

k = 2, 17

; Option C.

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