Question

If the roots of the quadratic equation $$ (a^2 + b^2) \, x^2 - 2 \, (bc + ad) \, x + (c^2 + d^2) = 0 $$ are equal, then: 

• A. \( \frac{a}{b} = \frac{c}{d} \)
• B. \( \frac{a}{c} + \frac{b}{d} = 0 \)
• C. \( \frac{a}{d} = \frac{b}{c} \)
• D. \( a + b = c + d \)

Hint:

For a quadratic equation \( Ax^2 + Bx + C = 0 \), the condition for equal roots is that the discriminant \( D = B^2 - 4AC = 0 \). Use this property to establish relationships between the coefficients.

Answer 1

The Correct Option is A

Step 1: Condition for Equal Roots For a quadratic equation of the form:\[Ax^2 + Bx + C = 0,\]the roots are equal if the discriminant \( D = B^2 - 4AC \) equals zero.
Step 2: Compute the Discriminant For the given equation:\[(a^2 + b^2) x^2 - 2 (bc + ad) x + (c^2 + d^2) = 0,\]the discriminant is:\[D = [-2 (bc + ad)]^2 - 4 (a^2 + b^2) (c^2 + d^2).\]Expanding this expression yields:\[D = 4 (bc + ad)^2 - 4 (a^2 + b^2) (c^2 + d^2).\]Factoring out the common term leads to:\[4 \left[ (bc + ad)^2 - (a^2 + b^2) (c^2 + d^2) \right] = 0.\]
Step 3: Solve for the Relationship Setting the expression in the brackets to zero:\[(bc + ad)^2 = (a^2 + b^2)(c^2 + d^2).\]Expanding both sides gives:\[b^2c^2 + 2abcd + a^2d^2 = a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2.\]Rearranging the terms to isolate the product of \(a, b, c, d\):\[2abcd = a^2c^2 + b^2d^2.\]Subtracting all terms to one side:\[2abcd - a^2c^2 - b^2d^2 = 0.\]Factoring this expression results in:\[(ad - bc)^2 = 0.\]This implies:\[ad = bc.\]Which can be rewritten as:\[\frac{a}{b} = \frac{c}{d}.\]
Step 4: Matching with the Options The obtained relationship, \( \frac{a}{b} = \frac{c}{d} \), corresponds to option (A).Final Answer: The correct condition is (A) \( \frac{a}{b} = \frac{c}{d} \).