Consider the equations of the circles:S_1 : x | vedclass

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(D) $S_1 : x^2 + y^2 + 24x - 10y + a = 0$. Center $C_1 = (-12, 5)$,radius $r_1 = \sqrt{144 + 25 - a} = \sqrt{169 - a}$.$S_2 : x^2 + y^2 = 36$. Center

Vedclass · Accepted Answer

If $a = 0$,then the number of common tangents to the circles $S_1 = 0$ and $S_2 = 0$ is $3$.

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(D) $S_1 : x^2 + y^2 + 24x - 10y + a = 0$. Center $C_1 = (-12, 5)$,radius $r_1 = \sqrt{144 + 25 - a} = \sqrt{169 - a}$.$S_2 : x^2 + y^2 = 36$. Center $C_2 = (0, 0)$,radius $r_2 = 6$.$(1)$ For a real circle,$r_1^2 \ge 0$ $\Rightarrow 169 - a \ge 0$ $\Rightarrow a \le 169$. Since $a$ is non-negative,$a \in \{0, 1, 2, \dots, 169\}$. Total values = $170$. Statement $A$ is correct.$(2)$ For no point in common,either $C_1C_2 > r_1 + r_2$ or $C_1C_2 $C_1C_2 = \sqrt{(-12)^2 + 5^2} = 13$.Case $1: 13 > \sqrt{169 - a} + 6$ $\Rightarrow 7 > \sqrt{169 - a}$ $\Rightarrow 49 > 169 - a$ $\Rightarrow a > 120$. Values: $121, \dots, 169$ ($49$ values).Case $2: 13 49$. Statement $B$ is correct.$(3)$ For orthogonal intersection,$2g_1g_2 + 2f_1f_2 = c_1 + c_2$.$2(12)(0) + 2(-5)(0) = a - 36$ $\Rightarrow 0 = a - 36$ $\Rightarrow a = 36$. Statement $C$ is correct.$(4)$ If $a = 0$,$r_1 = \sqrt{169} = 13$. $C_1C_2 = 13$. Since $|r_1 - r_2| < C_1C_2 < r_1 + r_2$ $(7 < 13 < 19)$,the circles intersect at two points. Number of common tangents is $2$. Statement $D$ is incorrect.

Consider the equations of the circles:
$S_1 : x^2 + y^2 + 24x - 10y + a = 0$
$S_2 : x^2 + y^2 = 36$
Which of the following statements is not correct?

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Consider the equations of the circles:$S_1 : x^2 + y^2 + 24x - 10y + a = 0$$S_2 : x^2 + y^2 = 36$Which of the following statements is not correct?