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UPSC NDA I 2026 Mathematics Paper

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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)

.

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Question : 60 of 120

Marks: +1, -0

If a circle is drawn through A, B and D, then what is the diameter of the circle ?

$3 \sqrt{10}$
$5 \sqrt{10}$
$3 \sqrt{12}$
$5 \sqrt{12}$

Solution:

Concept:

Find the circle passing through the concyclic points, then compute its diameter.

Explanation:

Let the circle be

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

.

For

A(0,2)

:

2v + w = -4

.

For

B(2,3)

:

2u + 3v + w = -13

.

For

C(4,5)

:

4u + 5v + w = -41

.

Solving gives

u = 5

,

v = -19

,

w = 34

.

So the circle is

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

.

For

D(0,k)

on it:

k^2 - 19k + 34 = 0

, giving

k = 17

.

Center is

\left(-\frac52,\frac{19}{2}\right)

.

Radius squared:

r^2 = \left(\frac52\right)^2 + \left(\frac{19}{2}\right)^2 - 34 = \frac{125}{2}

.

So

r = \frac{5\sqrt{10}}{2}

and diameter

= 2r = 5\sqrt{10}

.

Answer:

5\sqrt{10}

, i.e. option B.

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