CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

Let $$A, G$$ and $$H$$ be the AM, GM and HM of two positive numbers $$a$$ and $$b$$. The quadratic equation whose roots are $$A$$ and $$H$$ is


A
Ax2(A2+G2)x+AG2=0
loader
B
Ax2(A2+H2)x+AH2=0
loader
C
Hx2(H2+G2)x+HG2=0
loader
D
Gx2(H2+G2)x+GH2=0
loader

Solution

The correct options are
A $$ Ax^{2}-({A}^{2}+G^{2})x+{A}G^{2}=0$$
C $$ Hx^{2}-(H^{2}+G^{2})x+HG^{2}=0$$
As $$A,G,H$$ are A.M, G.M and H.M between $$a$$ and $$b$$.
Then $$A,G,H$$ is in G.P such that $${ G }^{ 2 }=AH$$
Now equation whose roots are $$A$$ and $$H$$ is
$${ x }^{ 2 }-\left( A+H \right) x+AH=0$$
Substituting $$H=\cfrac { { G }^{ 2 } }{ A } $$, we get
$${ x }^{ 2 }-\left( A+\cfrac { { G }^{ 2 } }{ A }  \right) x+ G^2 =0$$
$$\Rightarrow A{ x }^{ 2 }-\left( { A }^{ 2 }+{ G }^{ 2 } \right) x+A{ G }^{ 2 }=0$$

Hence, option A and similarly option C.

Maths

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
People also searched for
View More



footer-image