Quantities x, 8, y (x=/y) are in g.p. and x, y, -8 in a.p. Find the value of x and y.
first condition:
8/x = y/8
xy = 64
2nd condition:
y-x = -8-y
2y + 8 = x
in xy=64
(2y+8)(y) = 64
2y^2 + 8y - 64=0
y^2 + 4y - 32=0
(y+8)(y-4) = 0
y = -8 or y = 4
since xy= 64
if y = 4, x = 16
if y = -8 , x = -4
Thankyou sir
Let's start by solving the equations using the given information.
Given: x, 8, y are in a geometric progression (g.p.)
x, y, -8 are in an arithmetic progression (a.p.)
For the geometric progression:
We know that in a geometric progression, the ratio between consecutive terms is constant.
So, from x to 8, and then from 8 to y, the common ratio is the same.
That means we have the following equations:
8 / x = y / 8 .......(Equation 1)
y / 8 = -8 / y .......(Equation 2)
Now, let's solve Equation 1 for y:
8 / x = y / 8
64 = xy
Similarly, let's solve Equation 2 for y:
y / 8 = -8 / y
y^2 = -64
Since the value of y cannot be negative, there are no real solutions for y in this case.
Therefore, it is not possible to find the values of x and y given the conditions provided.
To solve this problem, we can start by analyzing the given information:
We know that the quantities x, 8, and y are in a geometric progression (g.p.).
We also know that the quantities x, y, and -8 are in an arithmetic progression (a.p.).
Let's use the formulas for g.p. and a.p. to set up equations and solve for x and y.
For the g.p., the common ratio (r) is calculated by dividing any term by its previous term. So, we have:
8 / x = y / 8
Cross-multiplying, we get:
64 = xy
Now, for the a.p., we can use the formula for the general term of an arithmetic progression:
Term = First term + (n - 1) * Common difference
In this case, we have:
x = First term
y = First term + 2 * Common difference
-8 = First term + 3 * Common difference
By subtracting the second equation from the third equation, we get:
x - y = -5 * Common difference
Substituting the value of Common difference from the third equation into the above equation, we have:
x - y = -5 * (-8 - First term)
Let's substitute the value of First term from the second equation into the above equation:
x - y = -5 * (-8 - y)
Simplifying, we get:
x - y = 40 + 5y
Rearranging the equation, we get:
x = 40 + 6y
Now, we can substitute this value of x into the g.p. equation we obtained earlier:
64 = (40 + 6y)y
Expanding and rearranging the equation, we get a quadratic equation:
6y^2 + 40y - 64 = 0
Now, we can solve this quadratic equation using the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / 2a
Applying the values a = 6, b = 40, and c = -64, we have:
y = (-40 ± √(40^2 - 4 * 6 * -64)) / (2 * 6)
After simplifying this equation, we get two possible values for y, which are -4 and 2/3.
Substituting these values of y into the equation x = 40 + 6y, we can find the corresponding values of x:
- If y = -4, then x = 40 + 6 (-4) = 16.
- If y = 2/3, then x = 40 + 6 (2/3) = 44/3.
Therefore, the possible pairs of values for x and y are (16, -4) and (44/3, 2/3).