Medium Practice — Set 1

Quick Recap — Straight Lines & Circles (JEE Application)

• Image of $(x_1,y_1)$ in $ax+by+c=0$: shift by $-\dfrac{2(ax_1+by_1+c)}{a^2+b^2}(a,b)$.
• Concurrency of three lines $\Leftrightarrow$ the $3\times3$ determinant of coefficients $=0$.
• Tangent length from $(x_1,y_1)$ to $S=0$ is $\sqrt{S_1}$; radical axis of $S=0,S'=0$ is $S-S'=0$.
• Number of common tangents depends on $d$ vs $r_1\pm r_2$; two circles are orthogonal iff $2g_1g_2+2f_1f_2=c_1+c_2$.