# If two circles intersect at n points, then which of the following is true ? a) n^2<=9, b) n^3<=17, c) 2n + 3 <=4, d) n - 3 = 1

Sep 3, 2016

None of the given are true if the two circles can be the same. If not, then a and b are both true.

#### Explanation:

Two circles may intersect at $0 , 1 , 2 ,$ or $\infty$ points (the last case being if the circles are the same). If the two circles are the same, then none of the given possibilities are true.

If we exclude the possibility of the two circles being the same, then we have $n \in \left\{0 , 1 , 2\right\}$, in which case both a and b are true, as $0 \le n \le 2$, meaning

${n}^{2} \le 4 < 9$, and ${n}^{3} \le 8 < 17$.

Note that c is only true when $n = 0$, and d is never true, as two circles cannot intersect at exactly $4$ points.