Question

If a circle passes through $P(0,1),Q(0,9)$ and touches the $x$ - axis, then which of the following statement (s) is / are TRUE ?

• A. centres of circles are $(\pm 3,\pm 5)$
• B. Equation of one of their direct common tangent is $y=10$
• C. Radii of both the circle is $5$.
• D. length of common chord of circle is $8$.

Answer 1

Answer: (B), (C), (D)

The correct options are
B Equation of one of their direct common tangent is $y=10$
C Radii of both the circle is $5$.
D length of common chord of circle is $8$.

Center will lie on the perpendicular bisector of the chord.

So, y-coordinate of center = 5.

$\to$ Radius of both the circle = 5.

Also, length of common chord = $9-1=8.$

Let the x coordinate of center be h.

Now,

Distance of center from (0,9) = r

$\sqrt{h^2+4^2}=5$

$\to h=\pm 3$

So, Centres of circle = $(\pm 3,5)$

Lastlly, it is clearly depicted from above figure that one of the direct common tangent of circles is y = 10.

Hence, (B) , (C) and (D) are correct answers.