1) write an equation for a circle with the center (2,0) and radius sqrt 11
2) Find the common ratio in the geometric sequence : 7, -28, 112 ....
3) In a recursive formula, each succeeding term is formulated from the next term. a) true b) false
4) find the four corners of the fundamental rectangle of the hyperbola, x^2/36 + y^2/49 = 1
5) which expression can be used to find the number of permutations of n distinct objects of which p are alike and q are alike?
a) n!/(n-r)!r!
b) n!/p!q!
c) n!/(n-r)!
d) none of these
can someone please help me on these I am stuck
1. C(h,k).
C(2,0).
(x-h)^2 + (y-k)^2 = r^2.
(x-2)^2 + (y-0)2 = 11.
Eq: (x-2)^2 + y^2 = 11.
1) To write the equation for a circle with center (h, k) and radius r, we use the equation:
(x - h)^2 + (y - k)^2 = r^2
In this case, the center is (2, 0) and the radius is √11. Substituting these values into the equation, we get:
(x - 2)^2 + (y - 0)^2 = (√11)^2
(x - 2)^2 + y^2 = 11
Therefore, the equation for the circle is (x - 2)^2 + y^2 = 11.
2) To find the common ratio in a geometric sequence, you divide any term by the previous term. In this sequence: 7, -28, 112, ...
To find the common ratio, divide the second term (-28) by the first term (7):
-28 / 7 = -4
To confirm, divide the third term (112) by the second term (-28):
112 / -28 = -4
Since both divisions yield the same result (-4), the common ratio in this geometric sequence is -4.
3) In a recursive formula, each succeeding term is formulated from the next term.
The statement is incorrect, so the correct answer is b) false.
A recursive formula is a way to define a sequence where each term is expressed in terms of the previous term(s). In other words, to find a term, you need to know the term(s) that come before it.
4) The equation you provided, x^2/36 + y^2/49 = 1, represents a hyperbola.
The general equation for a hyperbola in standard form is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Comparing this with the given equation, we can see that the center of the hyperbola is (h, k) = (0, 0) and the values of a and b can be found by taking the square root of the denominators:
a = √36 = 6
b = √49 = 7
Knowing the center and the values of a and b, we can find the four corners of the fundamental rectangle of the hyperbola by using the following points:
(±a, ±b)
Therefore, the four corners of the fundamental rectangle of the hyperbola are:
(6, 7), (-6, 7), (-6, -7), (6, -7).
5) The expression that can be used to find the number of permutations of n distinct objects, where p are alike and q are alike, is:
b) n! / (p! * q!)
In this expression, n is the total number of objects, p is the number of objects of one kind, and q is the number of objects of another kind.
The formula n! / (p! * q!) accounts for the fact that objects of each kind are identical, so their permutations are divided out to avoid overcounting.
Therefore, the correct expression to find the number of permutations is b) n! / (p! * q!).
1) The equation for a circle with center (h,k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
In this case, the center is (2, 0) and the radius is √11. Substituting these values, the equation for the circle is:
(x - 2)^2 + (y - 0)^2 = (√11)^2
(x - 2)^2 + y^2 = 11
2) To find the common ratio in a geometric sequence, we divide each term by the previous term. In this sequence:
7 ÷ -28 = -1/4
-28 ÷ 112 = -1/4
The common ratio is -1/4.
3) In a recursive formula, each succeeding term is formulated from the previous term, not the next term. Therefore, the statement is false.
4) The equation x^2/36 + y^2/49 = 1 represents a hyperbola.
To find the four corners of the fundamental rectangle of the hyperbola, we consider the major and minor axes. The major axis is along the x-axis, and its length is equal to 2a.
In this case, a = 6 (since the denominator of x^2 is 36). Therefore, the length of the major axis is 2a = 2(6) = 12.
The minor axis is along the y-axis, and its length is equal to 2b.
In this case, b = 7 (since the denominator of y^2 is 49). Therefore, the length of the minor axis is 2b = 2(7) = 14.
The four corners of the fundamental rectangle are:
(-a, -b) = (-6, -7)
(-a, b) = (-6, 7)
(a, -b) = (6, -7)
(a, b) = (6, 7)
5) The expression that can be used to find the number of permutations of n distinct objects of which p are alike and q are alike is:
c) n!/(n - p)!(q)!
So the correct answer is c) n!/(n-r)!