Question

Let $S_1$ and $S_2$ be two circles passing through $P(2,3)$ and touching the coordinate axes and $S$ be a circle having centre at point $P$ and radius equal to G.M. of radii of $S_1$ and $S_2$, then

• A. S=0 cuts y-axis but not the x-axis
• B. S=0 cuts y=x
• C. A.M. of radii of $S_1$ and $S_2$ is 5
• D. Radius of director circle of $S=0$ is $\sqrt{26}$

Answer 1

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

B, C, D

Answer 2

Circles touching the coordinate axes and passing through $P(2,3)$ have radii $r_1, r_2$ that are roots of $r^2 - 10r + 13 = 0$. From Vieta's formulas, $r_1+r_2=10$ and $r_1r_2=13$. The Arithmetic Mean (A.M.) of $r_1, r_2$ is $(r_1+r_2)/2 = 10/2 = 5$. Thus, Option C is correct. Circle $S$ has center $P(2,3)$ and radius $R = \sqrt{r_1r_2} = \sqrt{13}$. The equation of $S$ is $(x-2)^2 + (y-3)^2 = 13$, which simplifies to $x^2+y^2-4x-6y=0$. Intersection with y-axis ($x=0$) gives $y^2-6y=0 \implies y=0,6$. Intersection with x-axis ($y=0$) gives $x^2-4x=0 \implies x=0,4$. Circle $S$ cuts both axes, so Option A is false. Intersection with $y=x$ gives $2x^2-10x=0 \implies x=0,5$. Circle $S$ cuts $y=x$. Thus, Option B is correct. The director circle of $S$ (center $(2,3)$, radius $R=\sqrt{13}$) has radius $\sqrt{2}R = \sqrt{2}\sqrt{13} = \sqrt{26}$. Thus, Option D is correct.