If \frac{x}{y+z} = \frac{y}{z+x} = \frac{z}{ x+y}, then show that each ratio is \frac{1}{ 2} or -1.
WBBSE
Class X
Mathematics
Variation
03 Mark
2024
\frac{x}{y+z} = \frac{y}{z+x} = \frac{z}{x+y} = k (say)
⇒ x = k(y+z) — (i)
⇒ y = k(z+x) — (ii)
⇒ z = k(x+y) — (ii)
(i) + (ii) + (iii)
x + y + z = k(y+z) + k(z+x) + k(x+y)
or, x + y + z = k(y + z + z + x + x + y)
or, x + y + z = 2k(x + y + z)
or, (x + y + z) – 2k(x + y + z) = 0
or, (x + y + z) (1 – 2k) = 0
either, x + y + z = 0 or, 1 – 2k = 0 ⇒ k = \frac{1}{2}
Again,
| x + y + z = 0
⇒ y + z = – x ⇒ \frac{x}{y + z} = -1 |
x + y + z = 0
⇒ x + z = – y ⇒ \frac{y}{x + z} = -1 |
x + y + z = 0
⇒ x + y = – z ⇒ \frac{z}{x + y} = -1 |
\frac{x}{y + z} = \frac{y}{x + z} = \frac{z}{x + y} = -1 or \frac{1}{2} (Proved)
Related Questions
More WBBSE Hard level questions
1/12
Q.
If x is proportional to y and y is proportional to z, then show that...
A.
y ∝ z ⇒ y = k2z --- (i) x ∝ y...
Q.
If (√a + √b) ∝ (√a-√b) then show that (a + b) ∝ √(ab)
A.
(√a + √b) ∝ (√a-√b) ⇒ (√a + √b) = k (√a-√b)...
Q.
State True or False: If a ∝ b, b ∝ 1/c, and c ∝ d,...
A.
True Explanation: a ∝ b, b ∝ 1/c ⇒ a ∝ 1/c....
Q.
If x³ – 1/y³ ∝ x³ + 1/y³, then show that x ∝ 1/y.
A.
(x³ - ) ∝ (x³ + ) ⇒ x³ + ∝ x³...
Q.
If \frac{\text{a²}}{\text{b+c}} =\frac{\text{b²}}{\text{c+a}} = \frac{\text{c²}}{\text{a+b}} = 1, then show that \frac{1}{\text{1+a}} + \frac{1}{\text{1+b}} + \frac{1}{\text{1+c}}...
A.
= 1 ∴ a2 = b + c ; b2 = c...
Q.
If 15 farmers can cultivate 18 bighas of land in 5 days, determine by using...
A.
No. of farmers = A, no. of days = B & area...
Q.
If x ∝ √y and y = a², when x = 2a then find x²...
A.
Since x is proportional to √y, let x = k√y. Given y...
Q.
If x/2 = y/3 = z/4 then find the value of (3x + 4y +...
A.
Let the common ratio be k. Then, x = 2k, y =...
Q.
If x ∝ y, y ∝ z and z ∝ x, then find the relation...
A.
x ∝ y implies x = k1 y. y ∝ z implies...
Q.
If b ∝ a³ and a increase in the ratio of 2 : 3, then...
A.
b ∝ a³ or, b = ka³ b1 : b2 = ka1³...
Q.
State True or False: If a and b are in inverse variation, then a/b =...
A.
False Explanation: Inverse variation means a ∝ ⇒ ab = k (constant)....
Q.
If sin 17° = x\over y then show that sec 17° – sin 73° =...
A.
Given: sin 17° = Formula: sin² θ + cos² θ = 1...