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If xy = r, xz = r2, yz = r3, x + y + z = 13 and x2 + y2 + z2 = 91, then find z/y.
- a)3
- b)4
- c)7/3
- d)13/3
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
If xy = r, xz = r2, yz = r3, x + y + z = 13 and x2+ y2+ z2= 91, then f...
x + y + z = 13, on squaring both sides
⇒ x2 + y2 + z2 + 2(xy + yz + zx) = 169
⇒ 2(xy + yz + zx) = 169 – 91 = 78
⇒ (xy + yz + zx) = 39
⇒ (r + r2 + r3) = 39 = 3 + 9 + 27
⇒ r = 3
∴ z/y = xz/xy = r2/r = r = 3
⇒ x2 + y2 + z2 + 2(xy + yz + zx) = 169
⇒ 2(xy + yz + zx) = 169 – 91 = 78
⇒ (xy + yz + zx) = 39
⇒ (r + r2 + r3) = 39 = 3 + 9 + 27
⇒ r = 3
∴ z/y = xz/xy = r2/r = r = 3
Most Upvoted Answer
If xy = r, xz = r2, yz = r3, x + y + z = 13 and x2+ y2+ z2= 91, then f...
Given Information:
- xy = r
- xz = r^2
- yz = r^3
- x + y + z = 13
- x^2 + y^2 + z^2 = 91
Calculating z/y:
- From the given information, we know that z = r/y and x = r/z.
- Substituting these values into x + y + z = 13, we get r/z + r/y + z = 13.
- Multiplying through by zy, we get r*y + r*z + z^2 = 13yz.
- Substituting the values of r, r^2, and r^3 into the equation, we get y + z + z^2 = 13z.
- Rearranging the equation gives z^2 - 12z + y = 0.
- Since x^2 + y^2 + z^2 = 91, we can substitute the value of y from the previous equation.
- This gives x^2 + z^2 + (12z - z^2) = 91.
- Simplifying further gives x^2 + 12z = 91.
- Since x = r/z, we can substitute this into the equation to get (r/z)^2 + 12z = 91.
- This simplifies to r^2/z^2 + 12z = 91.
- Substituting the values of r^2 and r, we get z^3 + 12z = 91.
- Solving this cubic equation gives z = 3.
- Therefore, z/y = 3/y, and since y + z = 13, y = 10.
- Thus, z/y = 3/10 = 3.
- xy = r
- xz = r^2
- yz = r^3
- x + y + z = 13
- x^2 + y^2 + z^2 = 91
Calculating z/y:
- From the given information, we know that z = r/y and x = r/z.
- Substituting these values into x + y + z = 13, we get r/z + r/y + z = 13.
- Multiplying through by zy, we get r*y + r*z + z^2 = 13yz.
- Substituting the values of r, r^2, and r^3 into the equation, we get y + z + z^2 = 13z.
- Rearranging the equation gives z^2 - 12z + y = 0.
- Since x^2 + y^2 + z^2 = 91, we can substitute the value of y from the previous equation.
- This gives x^2 + z^2 + (12z - z^2) = 91.
- Simplifying further gives x^2 + 12z = 91.
- Since x = r/z, we can substitute this into the equation to get (r/z)^2 + 12z = 91.
- This simplifies to r^2/z^2 + 12z = 91.
- Substituting the values of r^2 and r, we get z^3 + 12z = 91.
- Solving this cubic equation gives z = 3.
- Therefore, z/y = 3/y, and since y + z = 13, y = 10.
- Thus, z/y = 3/10 = 3.
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