To prove that [tex]\(\frac{a}{b} = \frac{c}{d}\)[/tex] when the roots of the equation [tex]\((a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0\)[/tex] are equal, follow these steps:
1. Understand the general form of the quadratic equation:
The given equation can be written in the standard quadratic form [tex]\(Ax^2 + Bx + C = 0\)[/tex], where:
[tex]\[
A = a^2 + b^2, \quad B = -2(ac + bd), \quad C = c^2 + d^2.
\][/tex]
2. Condition for equal roots:
To have equal roots for the quadratic equation [tex]\(Ax^2 + Bx + C = 0\)[/tex], the discriminant [tex]\(\Delta\)[/tex] must be zero. The discriminant of a quadratic equation [tex]\(Ax^2 + Bx + C = 0\)[/tex] is given by:
[tex]\[
\Delta = B^2 - 4AC.
\][/tex]
Setting the discriminant to zero:
[tex]\[
B^2 - 4AC = 0.
\][/tex]
3. Substitute [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]:
Substitute the values of [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] into the discriminant equation:
[tex]\[
(-2(ac + bd))^2 - 4(a^2 + b^2)(c^2 + d^2) = 0.
\][/tex]
4. Simplify the equation:
Simplify the expression:
[tex]\[
4(ac + bd)^2 - 4(a^2 + b^2)(c^2 + d^2) = 0.
\][/tex]
Divide through by 4 to simplify further:
[tex]\[
(ac + bd)^2 - (a^2 + b^2)(c^2 + d^2) = 0.
\][/tex]
Expanding both sides:
[tex]\[
(ac + bd)^2 = (a^2 + b^2)(c^2 + d^2).
\][/tex]
Expand [tex]\((ac + bd)^2\)[/tex]:
[tex]\[
a^2c^2 + 2abcd + b^2d^2 = a^2c^2 + b^2c^2 + a^2d^2 + b^2d^2.
\][/tex]
5. Compare both sides:
Comparing the terms on both sides of the equation:
[tex]\[
2abcd = b^2c^2 + a^2d^2.
\][/tex]
Rearrange the terms to isolate [tex]\(abcd\)[/tex]:
[tex]\[
a^2d^2 + b^2c^2 - 2abcd = 0.
\][/tex]
Recognize this as a perfect square:
[tex]\[
(ad - bc)^2 = 0.
\][/tex]
Therefore:
[tex]\[
ad - bc = 0.
\][/tex]
6. Conclude the proof:
From [tex]\(ad - bc = 0\)[/tex], we obtain:
[tex]\[
ad = bc.
\][/tex]
Finally, dividing both sides by [tex]\(bd\)[/tex]:
[tex]\[
\frac{a}{b} = \frac{c}{d}.
\][/tex]
Thus, we have proven that if the roots of the equation [tex]\((a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0\)[/tex] are equal, then [tex]\(\frac{a}{b} = \frac{c}{d}\)[/tex].
