Answer:
28) E
31) D
32) B
26) E
Step-by-step explanation:
Question 28
Equation of a circle: (x - a)² + (y - b)² = r²
where (a, b) is the center of the circle, and r is the radius
Given:
- center = (-5, 5)
- radius = 3
⇒ (x + 5)² + (y - 5)² = 9
Question 31
Vertical Parabola Equation: [tex]y=\dfrac{1}{4p}(x-h)^2+k[/tex]
where
- vertex: (h, k)
- focus: (h, k + p)
- directrix: y = k - p
The vertex of the parabola is at equal distance between focus and the directrix.
If the focus is (9, 27) and the directrix is at y = 11,
then the y-coordinate of the vertex will be: (27 - 11)/2 + 11 = 19
So the vertex is (9, 19)
Also, y-coordinate of focus: 19 + p = 27 ⇒ p = 8
[tex]\implies f(x)=\frac{1}{32}(x-9)^2+19[/tex]
Question 32
Horizontal Parabola Equation: [tex]x=\dfrac{1}{4p}(y-k)^2+h[/tex]
where
- vertex: (h, k)
- focus: (h + p, k)
- directrix: x = h - p
The vertex of the parabola is at equal distance between focus and the directrix.
If the focus is (-1, 15) and the directrix is at x = -4,
then the x-coordinate of the vertex will be: (-1 + 4)/2 = -2.5
So the vertex is (-2.5, 15)
Also, -2.5 - p = -4 ⇒ p = 1.5
[tex]\implies x=\dfrac{1}{6}(y-15)^2-\dfrac52[/tex]
Question 26
[tex](x - 3)^2+(y+4)^2=6^2[/tex]
Domain: 3 - 6 ≤ x ≤ 3 + 6 ⇒ -3 ≤ x ≤ 9
Range: -4 - 6 ≤ y ≤ -4 + 6 ⇒ -10 ≤ y ≤ 2
Therefore, the only point that satisfies the above domain and range is
(-3, -4)
