If the roots of the equation, (a 2 +b 2 )x 2 -2(ac+bd)x+(c 2 +d 2 ) = 0 are equal the
prove that ad = bc.
Answers
Answered by
12
Answer:
ad = bc
Step-by-step explanation:
Given that :
- The roots of the quadratic equation, (a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0 are equal.
To prove :
- ad = bc
Proof :
We know that,
The standard form of a quadratic equation is ax² + bx + c = 0. Here,
- a = (a² + b²)
- b = - 2(ac + bd)
- c = (c² + d²)
It is given that the roots are equal, therefore discriminant is zero.
Discriminant = b² - 4ac = 0
⇒ b² - 4ac = 0
⇒ {- 2(ac + bd)}² - 4(a² + b²)(c² + d²) = 0
⇒ 4(ac + bd)² - 4(a² + b²)(c² + d²) = 0
⇒ 4 [ (ac + bd)² - (a² + b²)(c² + d²) ] = 0
⇒ (ac + bd)² - (a² + b²)(c² + d²) = 0
⇒ a²c² + b²d² + 2. ac. bd - a²c² - a²d² - b²c² - b²d² = 0
⇒ - a²d² - b²c² + 2abcd = 0
⇒ - (a²d² + b²c² - 2abcd) = 0
⇒ a²d² + b²c² - 2abcd = 0
⇒ (ad - bc)² = 0
⇒ ad - bc = 0
⇒ ad = bc
Hence, it is proved.
Answered by
10
- If roots are equal , then ∆ = 0
- Given equation,
- (a²+b²)x² - 2(ac+bd)x + (c²+d²) = 0
Calculating the value of ∆
- ∆ = (2(ac+bd))² - 4(a²+b²)(c²+d²) = 0
- 4(ac)²+4(bd)²+8acbd -4(ac)² -4(ad)² -4(bc)² -4(bd)² = 0
- 8abcd - 4(ad)² -4(bc)² = 0
- -4(ad-bc)² = 0
- ad - bc = 0
- ad = bc
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