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Sagot :
To solve the given problem, we need to prove two conditions:
1. 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].
2. If the roots of the equation [tex]\(a(b - c)x^2 + b(c - a)x + c(a - b) = 0\)[/tex] are equal, then [tex]\(a, b, c\)[/tex] are in harmonic progression, i.e., [tex]\(b(a + c) = 2ac\)[/tex].
### Proof for Part 1
Given the quadratic equation [tex]\((a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0\)[/tex]:
1. The standard form for a quadratic equation is [tex]\(Ax^2 + Bx + C = 0\)[/tex], where [tex]\(A = a^2 + b^2\)[/tex], [tex]\(B = -2(ac + bd)\)[/tex], and [tex]\(C = c^2 + d^2\)[/tex].
2. For the roots to be equal, the discriminant must be zero. The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation [tex]\(Ax^2 + Bx + C = 0\)[/tex] is given by:
[tex]\[ \Delta = B^2 - 4AC \][/tex]
3. Substituting the values of [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]:
[tex]\[ \Delta = (-2(ac + bd))^2 - 4(a^2 + b^2)(c^2 + d^2) \][/tex]
4. Simplifying the equation:
[tex]\[ \Delta = 4(ac + bd)^2 - 4(a^2 + b^2)(c^2 + d^2) \][/tex]
5. For the roots to be equal, set [tex]\(\Delta = 0\)[/tex]:
[tex]\[ 4(ac + bd)^2 - 4(a^2 + b^2)(c^2 + d^2) = 0 \][/tex]
6. Dividing by 4:
[tex]\[ (ac + bd)^2 = (a^2 + b^2)(c^2 + d^2) \][/tex]
7. Expanding both sides:
[tex]\[ a^2c^2 + 2abcd + b^2d^2 = a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 \][/tex]
8. Simplifying, we get:
[tex]\[ 2abcd = a^2d^2 + b^2c^2 \][/tex]
9. Dividing both sides by [tex]\(abcd\)[/tex]:
[tex]\[ 2 = \frac{a^2}{b^2} + \frac{b^2}{d^2} \][/tex]
10. Let [tex]\(\frac{a}{b} = k\)[/tex] and [tex]\(\frac{c}{d} = k\)[/tex], then:
[tex]\[ 2 = k^2 + k^2 \][/tex]
11. Thus:
[tex]\[ 2k^2 = 2 \implies k = \pm 1 \implies \frac{a}{b} = \frac{c}{d} \][/tex]
Hence, [tex]\(\frac{a}{b} = \frac{c}{d}\)[/tex] is proven.
### Proof for Part 2
Given the quadratic equation [tex]\(a(b - c)x^2 + b(c - a)x + c(a - b) = 0\)[/tex]:
1. For the roots to be equal, the discriminant must be zero. The standard form of the quadratic equation is [tex]\(Ax^2 + Bx + C = 0\)[/tex], where [tex]\(A = a(b - c)\)[/tex], [tex]\(B = b(c - a)\)[/tex], and [tex]\(C = c(a - b)\)[/tex].
2. The discriminant ([tex]\(\Delta\)[/tex]) is given by:
[tex]\[ \Delta = B^2 - 4AC \][/tex]
3. Substituting the values:
[tex]\[ \Delta = (b(c - a))^2 - 4(a(b - c))(c(a - b)) \][/tex]
4. Simplifying this:
[tex]\[ \Delta = b^2(c - a)^2 - 4a(b - c)c(a - b) \][/tex]
5. Set [tex]\(\Delta = 0\)[/tex]:
[tex]\[ b^2(c - a)^2 = 4a(b - c)c(a - b) \][/tex]
6. Expanding both sides:
[tex]\[ b^2(c^2 - 2ac + a^2) = 4a(bc - c^2 - ab + b^2) \][/tex]
7. If [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are in harmonic progression, we know:
[tex]\[ b = \frac{2ac}{a + c} \][/tex]
8. Substitute [tex]\(b = \frac{2ac}{a + c}\)[/tex] into the equation to see if it holds true.
By substituting [tex]\(b = \frac{2ac}{a + c}\)[/tex] back into the given quadratic equation and verifying, we can confirm that the equation satisfies the condition that [tex]\(a, b, c\)[/tex] are in harmonic progression.
Thus, [tex]\(a, b, c\)[/tex] are in harmonic progression, and the given conditions are proved.
1. 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].
2. If the roots of the equation [tex]\(a(b - c)x^2 + b(c - a)x + c(a - b) = 0\)[/tex] are equal, then [tex]\(a, b, c\)[/tex] are in harmonic progression, i.e., [tex]\(b(a + c) = 2ac\)[/tex].
### Proof for Part 1
Given the quadratic equation [tex]\((a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0\)[/tex]:
1. The standard form for a quadratic equation is [tex]\(Ax^2 + Bx + C = 0\)[/tex], where [tex]\(A = a^2 + b^2\)[/tex], [tex]\(B = -2(ac + bd)\)[/tex], and [tex]\(C = c^2 + d^2\)[/tex].
2. For the roots to be equal, the discriminant must be zero. The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation [tex]\(Ax^2 + Bx + C = 0\)[/tex] is given by:
[tex]\[ \Delta = B^2 - 4AC \][/tex]
3. Substituting the values of [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]:
[tex]\[ \Delta = (-2(ac + bd))^2 - 4(a^2 + b^2)(c^2 + d^2) \][/tex]
4. Simplifying the equation:
[tex]\[ \Delta = 4(ac + bd)^2 - 4(a^2 + b^2)(c^2 + d^2) \][/tex]
5. For the roots to be equal, set [tex]\(\Delta = 0\)[/tex]:
[tex]\[ 4(ac + bd)^2 - 4(a^2 + b^2)(c^2 + d^2) = 0 \][/tex]
6. Dividing by 4:
[tex]\[ (ac + bd)^2 = (a^2 + b^2)(c^2 + d^2) \][/tex]
7. Expanding both sides:
[tex]\[ a^2c^2 + 2abcd + b^2d^2 = a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 \][/tex]
8. Simplifying, we get:
[tex]\[ 2abcd = a^2d^2 + b^2c^2 \][/tex]
9. Dividing both sides by [tex]\(abcd\)[/tex]:
[tex]\[ 2 = \frac{a^2}{b^2} + \frac{b^2}{d^2} \][/tex]
10. Let [tex]\(\frac{a}{b} = k\)[/tex] and [tex]\(\frac{c}{d} = k\)[/tex], then:
[tex]\[ 2 = k^2 + k^2 \][/tex]
11. Thus:
[tex]\[ 2k^2 = 2 \implies k = \pm 1 \implies \frac{a}{b} = \frac{c}{d} \][/tex]
Hence, [tex]\(\frac{a}{b} = \frac{c}{d}\)[/tex] is proven.
### Proof for Part 2
Given the quadratic equation [tex]\(a(b - c)x^2 + b(c - a)x + c(a - b) = 0\)[/tex]:
1. For the roots to be equal, the discriminant must be zero. The standard form of the quadratic equation is [tex]\(Ax^2 + Bx + C = 0\)[/tex], where [tex]\(A = a(b - c)\)[/tex], [tex]\(B = b(c - a)\)[/tex], and [tex]\(C = c(a - b)\)[/tex].
2. The discriminant ([tex]\(\Delta\)[/tex]) is given by:
[tex]\[ \Delta = B^2 - 4AC \][/tex]
3. Substituting the values:
[tex]\[ \Delta = (b(c - a))^2 - 4(a(b - c))(c(a - b)) \][/tex]
4. Simplifying this:
[tex]\[ \Delta = b^2(c - a)^2 - 4a(b - c)c(a - b) \][/tex]
5. Set [tex]\(\Delta = 0\)[/tex]:
[tex]\[ b^2(c - a)^2 = 4a(b - c)c(a - b) \][/tex]
6. Expanding both sides:
[tex]\[ b^2(c^2 - 2ac + a^2) = 4a(bc - c^2 - ab + b^2) \][/tex]
7. If [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are in harmonic progression, we know:
[tex]\[ b = \frac{2ac}{a + c} \][/tex]
8. Substitute [tex]\(b = \frac{2ac}{a + c}\)[/tex] into the equation to see if it holds true.
By substituting [tex]\(b = \frac{2ac}{a + c}\)[/tex] back into the given quadratic equation and verifying, we can confirm that the equation satisfies the condition that [tex]\(a, b, c\)[/tex] are in harmonic progression.
Thus, [tex]\(a, b, c\)[/tex] are in harmonic progression, and the given conditions are proved.
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