Question

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Answer 1

$a)quad x^2+y^2-6x+8y=0\\(x-3)^2-9+(y+4)^2-16=0\\(x-3)^2+(y+4)^2=25\\okryznost; ,; ; centr; v; ; C(3;-4); ,; R=5\\b)quad 9x^2+16y^2=144\\ frac{9x^2}{144}+frac{16y^2}{144}=1\\frac{x^2}{16}+frac{y^2}{9}=1\\ellips; ,; ; centr; v; ; O(0;0); ,; a=sqrt{16}=4; ,; ; b=sqrt9=3\\c^2=a^2-b^2=16-9=7; ,; ; c=sqrt7\\F_1(c;0); ,; ; F_2(-c;0)\\F_1(sqrt7;0)quad -; ; pravuj; fokys\\c)quad pryamaya; l:; ; frac{x-x_1}{x_2-x_1}=frac{y-y_1}{y_2-y_1}$

$C(3;-4); ,; ; F_1(sqrt7;0)\\ frac{x-3}{sqrt7-3}=frac{y+4}{0+4}\\4(x-3)=(sqrt7-3)(y+4)\\4x-12=(sqrt7-3)y+4(sqrt7-3)\\y=frac{1}{sqrt7-3} cdot (4x-12-4(sqrt7-3))\\y=frac{4}{sqrt7-3}cdot (x-3-sqrt7+3)\\y=frac{4}{sqrt7-3}cdot x-frac{4sqrt7}{sqrt7-3}\\y=frac{4(sqrt7+3)}{7-9}cdot x- frac{4sqrt7(sqrt7+3)}{7-9} \\y=-2(sqrt7+3)cdot x+2sqrt7(sqrt7+3)\\y=-2(sqrt7+3)cdot x+14+6sqrt7$