Answer

Verified

381.6k+ views

**Hint:**Start by comparing with the standard quadratic equation and find out the discriminant value, and according to the condition of equal roots equate D to zero i.e. D=0, Find out the required condition by simplifying the relation.

**Complete step-by-step solution:**Given , $({a^2} + {b^2}){x^2} - 2(ac + bd)x + ({c^2} + {d^2}) = 0 \to (1)$

Step by step solution

We know for any quadratic equation $A{x^2} + Bx + C = 0$, equal roots are possible only when discriminant(D) = 0. Which can be found by the formula $D = {B^2} - 4AC$.

On comparing with equation 1, we get

$

A = ({a^2} + {b^2}) \\

B = - 2(ac + bd) \\

C = ({c^2} + {d^2}) $

Using the above concept we need to find discriminant (D), we get :

$ D = {\left[ { - 2(ac + bd)} \right]^2} - 4 \times ({a^2} + {b^2}) \times ({c^2} + {d^2}) \\

= 4({a^2}{c^2} + {b^2}{d^2} + 2ac \cdot bd) - 4({a^2}{c^2} + {b^2}{d^2} + {a^2}{d^2} + {b^2}{c^2}) $

Here, we used the formula ${(a + b)^2} = {a^2} + {b^2} + 2ab$

Now , $D = 0$

$\Rightarrow 4({a^2}{c^2} + {b^2}{d^2} + 2ac \cdot bd) - 4({a^2}{c^2} + {b^2}{d^2} + {a^2}{d^2} + {b^2}{c^2}) = 0 \\

\Rightarrow ({a^2}{c^2} + {b^2}{d^2} + 2ac \cdot bd) = ({a^2}{c^2} + {b^2}{d^2} + {a^2}{d^2} + {b^2}{c^2}) \\

\Rightarrow {a^2}{d^2} + {b^2}{c^2} - 2ac \cdot bd = 0 \\

\Rightarrow {(ad - bc)^2} = 0 \\

\Rightarrow ad - bc = 0 \\

\Rightarrow ad = bc $

**So, option B is the correct option.**

**Note:**For any quadratic equation $A{x^2} + Bx + C = 0$, there are three types of roots available which can only be determined after calculating Discriminant(D) by the formula or relation $D = {B^2} - 4AC$ . Now the three types of roots are as follows:

i). D>0 , Distinct and real roots exist and the roots are $\alpha = \dfrac{{ - B + \sqrt D }}{{2A}},\beta = \dfrac{{ - B - \sqrt D }}{{2A}}$

ii). D=0, Real and equal roots exist and the roots are $\alpha = \beta = \dfrac{{ - B}}{{2A}}$

iii). D<0, Imaginary roots exist.

For imaginary roots, there's a whole different chapter and concept known as “Complex numbers and roots”.

Recently Updated Pages

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

What happens when entropy reaches maximum class 11 chemistry JEE_Main

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Why is the adrenaline hormone called fight or flight class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between lanthanoids and actinoids class 12 chemistry CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Give 10 examples of unisexual and bisexual flowers

Open circulatory system is present in I Arthropods class 12 biology CBSE

Name the highest peak of the Indian Himalayas class 8 social science CBSE