Apotemi Yayinlari Analitik Geometri May 2026

Use ( x_0^2 + y_0^2 = 16 ): [ \left( \frac23(Y - 1) \right)^2 + \left( -\frac23(X + 2) \right)^2 = 16. ] [ \frac49 (Y - 1)^2 + \frac49 (X + 2)^2 = 16. ] Multiply by ( 9/4 ): [ (Y - 1)^2 + (X + 2)^2 = 36. ]

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) ). Apotemi Yayinlari Analitik Geometri

( |t_1 - t_2| = \frac\sqrt\Delta ), where ( \Delta = (-2m)^2 - 4(1+m^2)(-35) = 4m^2 + 140(1+m^2) = 4m^2 + 140 + 140m^2 = 144m^2 + 140 ). So ( |t_1 - t_2| = \frac\sqrt144m^2 + 1401+m^2 ). Thus [ \textArea(m) = 2m \cdot \frac\sqrt144m^2 + 1401+m^2. ] Use ( x_0^2 + y_0^2 = 16 ):

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. ] Express ( x_0, y_0 ) in terms

Cancel ( 1152u^2 ) both sides: ( 1712u + 560 = 1120u \implies 592u = -560 ) — impossible for ( u>0 ).

Area of triangle ( A(2,0), R_1, R_2 ): Use determinant formula: [ \textArea = \frac12 | x_A(y_1 - y_2) + x_1(y_2 - y_A) + x_2(y_A - y_1) |. ] Better: shift coordinates to simplify. Let ( u = x-2, v = y ) (translate so ( A ) at origin). Then ( A'=(0,0) ), ( R_i' = (t_i - 4, m t_i) ). Area = ( \frac12 | (t_1-4)(m t_2) - (t_2-4)(m t_1) | ) (since ( \frac12 |x_1 y_2 - x_2 y_1| ) in translated coords). Simplify: [ (t_1-4)m t_2 - (t_2-4)m t_1 = m[ t_1 t_2 - 4 t_2 - t_1 t_2 + 4 t_1 ] = m[ 4(t_1 - t_2) ]. ] So Area = ( \frac12 | 4m (t_1 - t_2) | = 2m |t_1 - t_2| ).

Rotation of ( Q ) about ( B(-2,0) ) by ( +90^\circ ). Vector from ( B ) to ( Q ): [ \vecBQ = Q - B = \left( \frac32x_0 - 1 + 2, \ \frac32y_0 - 0 \right) = \left( \frac32x_0 + 1, \ \frac32y_0 \right). ] Rotation by ( 90^\circ ) CCW: ( (u, v) \mapsto (-v, u) ). So [ \vecBR = \left( -\frac32y_0, \ \frac32x_0 + 1 \right). ] Thus [ R = B + \vecBR = \left( -2 - \frac32y_0, \ 0 + \frac32x_0 + 1 \right). ] Let ( R = (X, Y) ): [ X = -2 - \frac32y_0, \quad Y = 1 + \frac32x_0. ]