Question

A circle is drawn touching both the axes. The equation of a chord with P(3,2) as midpoint is x=3. If P lies one unit away from the centre of the circle, find the length of the chord.

• A. √3
• B. 2√3
• C. √5
• D. 2√5

Answer 1

The correct option is B

2√3

Its given that the circle touches both the axes. For drawing such a circle we need to know which quadrant its lying in. Its also given that midpoint of one of its chord is (3, 2). So we get that the circle is in first quadrant.

Lets draw the circle,

Now lets draw the given chord with equation x = 3 and midpoint (3, 2)

Since circle is touching both the axis , the x and y coordinate the centre of the circle will be equal to the radius of the cicle. So the centre will be of the form (a, a) since x coordinate = y coordinate = radius

Since x = 3 is parallel to y axis, midpoint of chord will be on the same horizontal line drawn through the centre,

i.e., y-coordinate of centre = y-coordinate of P

∴ a = 2 = radius

OP = XP - XO

= 3 - 2 = 1 unit

OQ = radius = 2 units

$\therefore PQ=\frac{lengthofchord}{2}=\sqrt{2^2-1}=\sqrt{3}$

$\therefore$Length of chord,$RQ=2\sqrt{3}$