Question

If the ratio of the roots of the equation \(x^2 - 2ax + b = 0\) is equal to that of the roots of the equation \(x^2 - 2cx + d = 0\), then:

• A. \(a^2 b = c^2 d\)
• B. \(a^2 c = b^2 d\)
• C. \(a^2 d = c^2 b\)
• D. \(d^2 b = c^2 a\)

Hint:

Write the roots as mk and k using the given ratio, express b/a^2 in terms of the ratio alone, and do the same for d/c^2; the two expressions must then be equal.

Answer 1

The Correct Option is C

Step 1: Set up the roots of the first equation using their ratio.
Say the two roots of \(x^2 - 2ax + b = 0\) are in the ratio \(m : 1\). Write them as \(mk\) and \(k\) for some number \(k\). By Vieta's formulas, the sum of the roots is \(2a\) and the product is \(b\), so:
\[ mk + k = 2a \quad \Rightarrow \quad k(m+1) = 2a \]
\[ mk \cdot k = b \quad \Rightarrow \quad mk^2 = b \]

Step 2: Eliminate k to connect a, b and m alone.
From the sum equation, \(k = \dfrac{2a}{m+1}\). Substitute this into \(b = mk^2\):
\[ b = m \left( \frac{2a}{m+1} \right)^2 = \frac{4a^2 m}{(m+1)^2} \]
Dividing both sides by \(a^2\) gives a clean ratio that depends only on \(m\), not on \(a\) or \(k\):
\[ \frac{b}{a^2} = \frac{4m}{(m+1)^2} \]

Step 3: Repeat the same steps for the second equation.
The roots of \(x^2 - 2cx + d = 0\) have the same ratio \(m : 1\) (this is given), so writing them as \(mk'\) and \(k'\) for a different scale factor \(k'\), the identical working gives:
\[ \frac{d}{c^2} = \frac{4m}{(m+1)^2} \]
Notice the right-hand side is exactly the same expression as in Step 2, because it depends only on the common ratio \(m\), not on which equation we started from.

Step 4: Equate the two ratios and simplify.
Since both \(\dfrac{b}{a^2}\) and \(\dfrac{d}{c^2}\) equal the same quantity \(\dfrac{4m}{(m+1)^2}\), they must equal each other:
\[ \frac{b}{a^2} = \frac{d}{c^2} \]
Cross-multiplying gives \(b c^2 = d a^2\), which is the same as \(a^2 d = c^2 b\).
Options (A), (B) and (D) each pair up \(a, b, c, d\) differently (for instance option (A) puts \(b\) with \(c^2\) instead of matching each variable with its own equation's coefficients), so they do not match the relation that actually falls out of the algebra.

Final Answer:
The correct relation is \(a^2 d = c^2 b\), matching option (C). \[ \boxed{a^2 d = c^2 b} \]