# Question 3:

If the equation (1 + m²)x² + 2 mcx + c² – a² = 0 has equal roots then show that c² = a²(1 + m²).

[CBSE 2017]

## Solution:

(1 + m²) x² + 2mcx + c² – a² = 0

On comparing it with A x² + B x + C =0, we get

A = 1 + m², B = 2mc, C = c² – a²

The roots of given equation are equal, then

Discriminant, D = 0

∴  B² – 4AC = 0

(2mc)² – 4× (1 + m²) × (c² – a²) = 0

⇒ 4m²c² – 4(c² + c²m² – a² – a²m²) = 0

⇒ 4m²c² – 4c² – 4c²m² + 4a² + 4a²m² = 0

⇒  4a²m² + 4a² – 4c² = 0

⇒  a²m² + a² – c² = 0

⇒ c² = m²a² + a²

⇒  c²  =  a²(1 + m²)

Hence, proved

# Question:1

If Ritu were younger by 5 years than what she really is, then the square of her age would have been 11 more than five times her present age. What is her present age ?                   [CBSE Term-2 SQP 2022]

# Question 2:

The sum of areas of two squares is 157 m². If the sum of their perimeters is 68 m, find the sides of the two squares.                     [CBSE 2019]

# Question 4:

A train covers a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, it would have taken 3 hours more to cover the same distance. Find the original speed of the train.                [CBSE    2020]