Problem 1:
Equation of an ellipse is:
[tex]\[\frac{{{{(x - h)}^2}}}{a^2} + \frac{{{{(y - k)}^2}}}{b^2} = 1\][/tex]
[tex]\[\begin{array}{l}4{x^2} + {y^2} - 8x + 4y + 4 = 0\\4{x^2} - 8x + {y^2} + 4y + 4 = 0\\4({x^2} - 2x + {1^2}) + ({y^2} + 4y + {2^2}) = - 4 + 4({1^2}) + {2^2}\\4{(x - 1)^2} + {(y + 2)^2} = - 4 + 4 + 4\\4{(x - 1)^2} + {(y + 2)^2} = 4\\{(x - 1)^2} + \frac{{{{(y + 2)}^2}}}{4} = 1\end{array}\][/tex]
Problem 2:
[tex]\[\begin{array}{l}21{x^2} - 4{y^2} + 84x - 24y = 36\\21{x^2} + 84x - 4{y^2} - 24y = 36\\21({x^2} + 4x + {2^2}) - 4({y^2} + 6y + {3^2}) = 36 + 21({2^2}) - 4({3^2})\\21{(x + 2)^2} - 4{(y + 3)^2} = 36 + 84 - 36\\21{(x + 2)^2} - 4{(y + 3)^2} = 84\\\frac{{{{(x + 2)}^2}}}{4} - \frac{{{{(y + 3)}^2}}}{{21}} = 1\end{array}\][/tex]
The equation formed is a hyperbola
[tex]\[\frac{{{{(x - h)}^2}}}{a^2} - \frac{{{{(y - k)}^2}}}{b^2} = 1\][/tex]
c²= a² + b²
c² = 4 + 21
c² = 25
c = √25
c = 5
Center(h, k)
Center(-2, -3)
Focus(h + c, k), (h - c, k)
Focus(-2 + 5, -3), (-2 - 5, -3)
Focus(3, -3), (-7, -3)
Co-vertices(h, b - k), (h, b + k)
Co-vertices(-2, √21 - 3), (-2, - √21 - 3)
Co-vertices(-2, 1.58257569), (-2, -7.58257569)
Equation for Circle 1
where Focus(3, -3) of a hyperbola is the center of a circle
Center(3, -3) and passes through the farther vertex of a hyperbola (-2, 1.583)
get the radius of the circle by applying the distance formula
[tex]\[\begin{array}{l}r = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} \\\\r = \sqrt {{{( - 2 - 3)}^2} + {{(1.583 - ( - 3))}^2}} \\\\r = \sqrt {{{( - 5)}^2} + {{(4.583)}^2}} \\\\r = \sqrt {25 + 21} \\\\r = \sqrt {46} \end{array}\][/tex]
equation of the circle 1 is:
x² + y² = r²
(x - h)² + (y - k)² = r²
(x - 3)² + (y + 3)² = 46
Equation for Circle 2
where Focus(-7, -3) of a hyperbola is the center of a circle
Center(-7, -3) and passes through the farther vertex of a hyperbola (-2, -7.583)
[tex]\[\begin{array}{l}r = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} \\\\r = \sqrt {{{( - 2 - ( - 7))}^2} + {{( - 7.583 - ( - 3))}^2}} \\\\r = \sqrt {{{(5)}^2} + {{( - 4.583)}^2}} \\\\r = \sqrt {25 + 21} \\\\r = \sqrt {46} \end{array}\][/tex]
equation of the circle 2 is:
x² + y² = r²
(x - h)² + (y - k)² = r²
(x +7)² + (y + 3)² = 46
Answer:
Equation of Circle 1
(x - 3)² + (y + 3)² = 46
Equation of Circle 2
(x + 7)² + (y + 3)² = 46
Problem 3:
y² + 12x +6y = 39
(y - k)² = 4a(x - h)
y² + 6y + 3² = 39 - 12x + 3²
(y + 3)² = 48 - 12x
(y + 3)² = -12(x - 4)
V(h, k)
V(4, -3)
4a = -12
a = -3
Vertex of parabola is the vertex of the hyperbola
V(4, -3)
X = x + 4
X = 3 + 4
directrix: x = 7
conjugate axis of the hyperbola is on the directrix of a parabola
Conjugate axis which is at
Center (7, - 3)
Focus(a , 0)
X = x - 4 ; Y = y + 3
x = X + 4 ; y = Y - 3
x = -3 + 4 = 2 ; y = 0 - 3 = -3
Focus (1, -3)
Focus of parabola is the foci of the hyperbola
Focus (1, -3)
Equation of hyperbola
[tex]\[\frac{{{{(x - h)}^2}}}{a^2} - \frac{{{{(y - k)}^2}}}{b^2} = 1\][/tex]
h = 7
k = -3
a = 3
b = √27
c = 6
Answer:
[tex]\[\frac{{{{(x - 7)}^2}}}{9} - \frac{{{{(y + 3)}^2}}}{{27}} = 1\][/tex]
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