Question

what is the condition that the equation (m^2+n^2)x^2-2(mp+nq)x + p^2+q^2=0 has equal roots​

Answer 1

Answer:

m/n = p/q

Step-by-step explanation:

What is the condition that the equation (m^2+n^2)x^2-2(mp+nq)x + p^2+q^2=0 has equal roots​

(m² + n²)x²  - 2(mp + nq)x  + (p² + q²) = 0

for ax² + bx + c = 0

To have equal roots

b² = 4ac

here a = m² + n²  , b = - 2(mp + nq)   c = p² + q²

=> ( - 2(mp + nq))² = 4 (m² + n²)(p² + q²)

=> 4 (m²p² + n²q² + 2mpnq) = 4(m²p² + n²q² + m²q² + n²p²)

Cancelling 4 from both sides

=> m²p² + n²q² + 2mpnq = m²p² + n²q² + m²q² + n²p²

=> m²q² + n²p² - 2mpnq = 0

=> (mq - np)² = 0

=> mq = np

=> m/n = p/q

Answer 2

Answer:

m / n = p / q

Step-by-step explanation:

Given :

p ( x ) = ( m² + n² ) x² - 2 ( m p + n q ) x + p² + q² = 0

We know for equal and real root :

D = 0

i.e. b² - 4 a c = 0

( - 2 ( m p + n q ) )² - 4 ( m² + n² ) ( p² + q² ) = 0

4 ( ( m p + n q ) )² - 4 ( m² + n² ) ( p² + q² ) = 0

4 [ ( m p + n q )² - ( m² + n² ) ( p² + q² ) ] = 0-

( m p + n q )² - ( m² + n² ) ( p² + q² ) = 0

m² p² + n² q² + 2 m p n q - m² p² - m² q² - n² p² - n² q² = 0

2 m p n q -  m² q² - n² p² = 0

m² q² + n² p² - 2 m q n p = 0

( m q - n p )² = 0

m q - n p = 0

= > m q = n p

= > m / n = p / q

Therefore , the condition is m / n is equal to p / q.