1) To find the center of a circle given its equation, we need to rewrite the equation in the standard form (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center and r is the radius.
Given equation: 4x^2 + 4y^2 - 8x + 2y - 7 = 0
Rewrite it as: 4(x^2 - 2x) + 4(y^2 + 0.5y) = 7
Complete the square for x: 4(x^2 - 2x + 1) + 4(y^2 + 0.5y) = 7 + 4(1)
Simplify: 4(x - 1)^2 + 4(y + 0.25)^2 = 11
Divide by 4: (x - 1)^2 + (y + 0.25)^2 = 11/4
Comparing this with the standard form, we have:
Center at (1, -0.25)
2)
a)
Given that the common ratio of the GP is 3:
i) Let the first term of the AP be a and the common difference be d.
From the information given, we have:
14 = a + a = 2a, so a = 7
34 = a + d + a*3 = 7 + d + 21 --> d = 6
ii) The sixth term of the sequence:
AP: 7, 13, 19, 25, 31, 37
GP: 7, 21, 63, 189, 567, 1701
Adding corresponding terms: 14, 34, 78, 214, 598, 1738
Sum of the first six terms:
7 + 13 + 19 + 25 + 31 + 37 + 7 + 21 + 63 + 189 + 567 + 1701 = 2670
b) Given that the second term of the GP is 24 and the third term is 12(x + 1):
From the GP, we have:
24 = a*r
12(x + 1) = a*r^2
Solve for a and r:
24 = a*r
12(x + 1) = a*r^2
12(x + 1) = 24*r
x + 1 = 2r
Given that the sum of the first three terms is 76:
a + ar + ar^2 = 76
Substitute for a and r:
24 + 24r + 24r^2 = 76
24 + 24(2r) + 24(2r)^2 = 76
24 + 48r + 96r^2 = 76
96r^2 + 48r - 52 = 0
Divide by 4: 24r^2 + 12r - 13 = 0
Solve for r: r = 1, -13/24
If r = 1:
24 = a
a = 24
If r = -13/24:
24 = -13/24*a
a = -576
Therefore, the first term is either 24 or -576.
1)The equation of a circle is given by: 4x^2+4y^2-8x+2y-7=0.Determine the coordinates of the centre of the circle.
2)A sequence is formed by adding corresponding terms of an AP and GP.The first terms of the sequence formed are 14,34 and 78 respectively.
a)Given that the common ratio of G.P is 3;
i)Find the first term of the AP and GP and the common difference of the AP.
ii)Find the sixth term and the sum of the first six terms of the sequence.
b)The second and third terms of a geometric sequence progression are 24 and 12(x+1) respectively. Find the whole number value of x and hence.The first term given the sum of the first term given the sum of the first three terms of the progression is 76
1 answer
1 answer