Question

In the system of equations $\dfrac{x}{a} = \dfrac{y}{b}$, $ax + by = a^2 + b^2$, value of y will be

• A. $a$
• B. $ab$
• C. $b$
• D. $\dfrac{b}{a}$

Hint:

When equations involve ratios like $\dfrac{x}{a} = \dfrac{y}{b}$, always express one variable in terms of the other before substitution.

Answer 1

The Correct Option is D

Step 1: Given equations.
We are given:
1. $\dfrac{x}{a} = \dfrac{y}{b}$
2. $ax + by = a^2 + b^2$
Step 2: Express x in terms of y.
From $\dfrac{x}{a} = \dfrac{y}{b}$, we get $x = \dfrac{a}{b}y$.
Step 3: Substitute in second equation.
Substitute $x = \dfrac{a}{b}y$ into $ax + by = a^2 + b^2$:
\[ a\left(\dfrac{a}{b}y\right) + by = a^2 + b^2 \] \[ \dfrac{a^2 y}{b} + by = a^2 + b^2 \]
Step 4: Simplify.
Multiply both sides by $b$:
\[ a^2 y + b^2 y = b(a^2 + b^2) \] \[ y(a^2 + b^2) = b(a^2 + b^2) \]
Step 5: Divide both sides by $(a^2 + b^2)$.
\[ y = b \] Wait, check — but that contradicts the equation. Let's simplify again properly. Actually, after dividing both sides by $(a^2 + b^2)$, we see: \[ y = b \] But that’s not right — let's check carefully: \[ a^2y + b^2y = b(a^2 + b^2) \Rightarrow y(a^2 + b^2) = b(a^2 + b^2) \Rightarrow y = b \] Thus, the correct value is indeed $y = b$. Correction: Option (C) $b$ is correct.