If the equation (a^2+b^2 ) x^2-2(a c+b d) x+ (c^2+d^2 )=0 has equ… - Vidyadip

MCQ

If the equation $\left(a^2+b^2\right) x^2-2(a c+b d) x+\left(c^2+d^2\right)=0 $ has equal roots, then:

A
$\text{ab}=\text{cd}$
✓
$\text{ad}=\text{bc}$
C
$\text{ad}=\sqrt{\text{bc}}$
D
$\text{ab}=\sqrt{\text{cd}}$

✓

Answer

Correct option: B.

$\text{ad}=\text{bc}$

In the equation
$ \Rightarrow\left(a^2+b^2\right) x^2-2(a c+b d) x+\left(c^2+d^2\right)=0 $
$ \Rightarrow D=B^2-4 A C $
$ \Rightarrow D=[-2(a c+b d)]^2-4\left(a^2+b^2\right)\left(c^2+d^2\right) $
$ \Rightarrow D=4\left[a^2 c^2+b 2 d^2+2 a b c d\right]-4\left[a^2 c^2+a^2 d^2+b^2 c^2+b^2 d^2\right] $
$ \Rightarrow D=4 a^2 c^2+4 b^2 d^2+8 a b c d-4 a^2 c^2-4 a^2 d^2-4 b^2 c^2-4 b^2 d^2 $
$ \Rightarrow D=8 a b c d-4 a^2 d 2-4 b^2 c^2 $
$ \Rightarrow D=-4\left[a^2 d^2+b^2 c^2-2 a b c d\right] $
$ \Rightarrow D=-4(a d-b c)^2$
$\because$ Roots are equal
$\therefore D = 0$
$⇒ -4(ad - bc)^2= 0$
$⇒ ad - bc = 0$
$⇒ ad = bc$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Quadratic Equations→Quadratic Equations — all question types→Maths STD 10 question papers→CBSE Board STD 10 question papers→CBSE Board question paper generator→

Similar questions

If $\left(a^2+b^2\right) x^2+2(a b+b d) x+c^2+d^2=0$ has no real roots, then :

From the letters of the word $"\text{MOBILE}",$ a letter is selected. The probability that the letter is a vowel, is :

A ladder $15m$ long just reaches the top of a vertical wall. If the ladder makes an angle of $60^\circ$ with the wall, then the height of the wall is :

Choose the correct answer from the given four options : Twelve solid spheres of the same size are made by melting a solid metallic cylinder of base diameter $2\ cm$ and height $16\ cm$. The diameter of each sphere is :

If $2$ is a root of the equation $x^2+b x+12=0$ and the equation $x^2+b x+q=0$ has equal roots, then $q=$

If 3 is the least prime factor of $m$ and 5 is the least prime factor of $n$, then the least prime factor of $(m+n)$ is

Cards, each marked with one of the numbers $\{6, 7, 8, ..., 15\}$ are placed in a box and mixed thoroughly. One card is drawn at random from the box. What is the probability of getting a card with number less than $10?$

A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. The ratio of the volume of the smaller cone to the whole cone is:

$\triangle\text{ABC}\sim\triangle\text{DEF}$ such that AB = 9.1cm and DE = 6.5cm. If the perimeter of $\triangle\text{DEF}$ is 25cm, what is the perimeter of $\triangle\text{ABC}?$

A kite is flying at a height of $30\ m$ from the ground. The length of string from the kite to the ground is $60
\ m$ . Assuming that there is no slack in the string, the angle of elevation of the kite at the ground is