If two circles $(x-1)^{2}+(y-3)^{2}=a^{2}$ and $x^{2}+y^{2}-8x+2y+8=0$ intersect in two distinct points, then :

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To determine why Option 1 ($2 < a < 8$) is the correct answer, we need to analyze the conditions under which the two given circles intersect at two distinct points. ### Explanation: 1. **Equation of the First Circle:** The first circle is given by $(x-1)^2 + (y-3)^2 = a^2$. This circle has its center at $(1, 3)$ and radius $a$. 2. **Equation of the Second Circle:** The second circle is given by $x^2 + y^2 - 8x + 2y + 8 = 0$. We can rewrite this equation in standard form by completing the square: - For $x$: $x^2 - 8x$ can be rewritten as $(x-4)^2 - 16$. - For $y$: $y^2 + 2y$ can be rewritten as $(y+1)^2 - 1$. Substituting these into the equation, we get: $$ (x-4)^2 - 16 + (y+1)^2 - 1 + 8 = 0 \implies (x-4)^2 + (y+1)^2 = 9 $$ This represents a circle with center $(4, -1)$ and radius $3$. 3. **Condition for Intersection:** For the two circles to intersect at two distinct points, the distance between their centers must be less than the sum of their radii and greater than the absolute difference of their radii. - **Distance between centers:** The distance $d$ between the centers $(1, 3)$ and $(4, -1)$ is calculated as: $$ d = \sqrt{(4-1)^2 + (-1-3)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$ - **Sum of radii:** The sum of the radii is $a + 3$. - **Absolute difference of radii:** The absolute difference of the radii is $|a - 3|$. For intersection at two distinct points: $$ |a - 3| < 5 < a + 3 $$ This results in two inequalities: - $a - 3 < 5$ which simplifies to $a < 8$. - $5 < a + 3$ which simplifies to $a > 2$. Combining these inequalities, we get $2 < a < 8$. ### Justification for Other Options: - **Option 2: $a = 2$** - If $a = 2$, the absolute difference of the radii would be $|2 - 3| = 1$, which is less than the distance between the centers (5). Therefore, the circles would not intersect at two distinct points. - **Option 3: $a < 2$** - If $a < 2$, the absolute difference of the radii would be greater than the distance between the centers, which would mean the circles do not intersect at all. - **Option 4: $a > 2$** - While $a > 2$ is a necessary condition for intersection, it is not sufficient by itself. The upper bound $a < 8$ is also required to ensure the circles intersect at two distinct points. Thus, the correct answer is Option 1: $2 < a < 8$, as it satisfies the condition for the circles to intersect at two distinct points.