: Center ( (1, -2) ), ( a^2 = 25 \implies a = 5 ), ( b^2 = 9 \implies b = 3 ). Vertices: ( (1 \pm 5, -2) ) → ( (6, -2) ) and ( (-4, -2) ). ( c = \sqrta^2 - b^2 = \sqrt25 - 9 = 4 ). Foci: ( (1 \pm 4, -2) ) → ( (5, -2) ) and ( (-3, -2) ). 10. Hyperbola (Horizontal Transverse Axis) Equation : [ \frac(x - h)^2a^2 - \frac(y - k)^2b^2 = 1 ] Center ( (h, k) ), vertices ( (h \pm a, k) ), foci ( (h \pm c, k) ), ( c^2 = a^2 + b^2 ). ✅ Solved Exercise 10 Find center, vertices, foci of ( \frac(x - 2)^216 - \frac(y + 1)^29 = 1 ).
: [ M_x = \frac-2 + 62 = \frac42 = 2, \quad M_y = \frac4 + (-8)2 = \frac-42 = -2 ] geometria analitica conamat ejercicios resueltos
: ( (3, 9) ) and ( (-1, 1) ) 8. Parabola Vertex, Focus, Directrix Vertical parabola : ( (x - h)^2 = 4p(y - k) ) Vertex ( (h, k) ), focus ( (h, k + p) ), directrix ( y = k - p ). ✅ Solved Exercise 8 Find vertex, focus, directrix of ( y = 2x^2 - 8x + 5 ). : Center ( (1, -2) ), ( a^2
: [ (x - 3)^2 + (y + 2)^2 = 16 ]
: [ y - 5 = -3(x - 2) \implies y - 5 = -3x + 6 \implies y = -3x + 11 ] Foci: ( (1 \pm 4, -2) ) → ( (5, -2) ) and ( (-3, -2) )
: Group ( x ) and ( y ) terms: [ (x^2 - 6x) + (y^2 + 4y) = 3 ] Complete the square: [ (x^2 - 6x + 9) + (y^2 + 4y + 4) = 3 + 9 + 4 ] [ (x - 3)^2 + (y + 2)^2 = 16 ] Center ( C(3, -2) ), radius ( r = 4 ). 7. Intersection of a Line and a Parabola ✅ Solved Exercise 7 Find intersection points between ( y = x^2 ) and ( y = 2x + 3 ).