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 \( D = B^2 - 4AC \) is zero.
Step 2: Compute the Discriminant Given the 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: \[ D = 4 (bc + ad)^2 - 4 (a^2 + b^2) (c^2 + d^2). \] Factoring out the common term: \[ 4 \left[ (bc + ad)^2 - (a^2 + b^2) (c^2 + d^2) \right] = 0. \]
Step 3: Solve for the Relationship \[ (bc + ad)^2 = (a^2 + b^2)(c^2 + d^2). \] Expanding both sides: \[ b^2c^2 + 2abcd + a^2d^2 = a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2. \] Rearranging: \[ 2abcd = a^2c^2 + b^2d^2. \] \[ 2abcd - a^2c^2 - b^2d^2 = 0. \] Factoring: \[ (ad - bc)^2 = 0. \] \[ ad = bc. \] \[ \frac{a}{b} = \frac{c}{d}. \]
Step 4: Matching with the Options The derived equation matches option (A): \( \frac{a}{b} = \frac{c}{d} \). Final Answer: The correct condition is (A) \( \frac{a}{b} = \frac{c}{d} \).