Apotemi Yayinlari Analitik Geometri -

Given typical contest style, maybe I made algebra slip. But this derivation shows area→0 as m→0. So possibly intended: line through B and tangent to circle? No, that yields one intersection. Hmm.

Mistake? Let’s recheck derivative carefully. Apotemi Yayinlari Analitik Geometri

Equate: ( 144u^3 + 358u^2 + 284u + 70 = 144u^3 + 284u^2 + 140u ). Cancel ( 144u^3 ): ( 358u^2 + 284u + 70 = 284u^2 + 140u ) ( (358-284)u^2 + (284-140)u + 70 = 0 ) ( 74u^2 + 144u + 70 = 0 ) Divide 2: ( 37u^2 + 72u + 35 = 0 ). Given typical contest style, maybe I made algebra slip

Express ( x_0, y_0 ) in terms of ( X, Y ): From ( X ): ( \frac32y_0 = -X - 2 ) ⇒ ( y_0 = -\frac23(X + 2) ). From ( Y ): ( \frac32x_0 = Y - 1 ) ⇒ ( x_0 = \frac23(Y - 1) ). No, that yields one intersection

[ \text(a) (x+2)^2+(y-1)^2=36 \quad \text(b) Circle, center (-2,1),\ r=6 \quad \text(c) \inf \text area =0 \text as m\to 0^+ ]

Thus final: minimal area 0 as m→0, but triangle degenerates. For non-degenerate, no minimum, but if they ask for minimizing area among non-degenerate, it's arbitrarily small.

That means ( h'(u) ) never zero for ( u>0 ) — so minimum at boundary ( u\to 0^+ ) or ( u\to\infty ). Check: As ( u\to 0^+ ), ( h(u) \sim 140u / 1 \to 0 ). As ( u\to\infty ), ( h(u) \sim 144u^2 / u^2 = 144 ). So ( h(u) ) increases from 0 to 144. So minimal area → 0 as ( m\to 0^+ ). But slope ( m>0 ), line through ( B(-2,0) ) — as ( m\to 0 ), line is horizontal ( y=0 ), intersects circle at two points symmetric about center’s vertical line? Wait, ( m=0 ) gives ( y=0 ), circle: ( (x+2)^2 + 1 = 36 ) ⇒ ( (x+2)^2 = 35 ) ⇒ two intersections. Then area formula: ( A=2m|t_1-t_2| ) with ( m=0 ) → area 0? But triangle degenerates? Yes, all points on x-axis: ( A(2,0) ) and ( R_1,R_2 ) on x-axis → collinear → area 0. But ( m>0 ) strictly? Problem says ( m>0 ), so infimum is 0 but not attained. Likely they expect answer for minimal positive area? Then no min, only infimum.