Help me.

1.The sum of the roots of the equation \(\frac{1}{x+a}+\frac{1}{x+b}=\frac{1}{c}\) is zero.Prove that the product of the roots is \(-\frac{1}{2}(a^2+b^2)\).

2.If the roots of the equation \(p(q-r)x^2+q(r-p)x+r(p-q)=0\) be equal , then show that \(\frac{1}{p}+\frac{1}{r}=\frac{2}{q}\).

3.Find the condition that one root of \(ax^2+bx+c=0\) shall be \('n'\) times the other.

4.If \(Sinx + Siny=a\) and \(Cosx + Cosy=b\),Then find the roots of equation \((a^2+b^2+2b)t^2-4at+a^2+b^2-2b=0\)

Note by Abhay Kumar
2 years, 11 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

A - 3. Let the one root of given equation is \(\beta\), then the other root must be n\(\beta\).
Using Vieta's formula, Sum of roots = \(\beta + n\beta = -\dfrac{b}{a} \Rightarrow \color{red}{\beta} = -\dfrac{b}{a(1 + n)}\).
Product of roots = \(n\color{red}{\beta}^2 = \dfrac{c}{a}\).
\( \Rightarrow n\left(\dfrac{-b}{a(1 + n)}\right)^2 = \dfrac{c}{a}\).
\( \Rightarrow nb^2 = ac + acn^2 + 2anc\).
\( \Rightarrow acn^2 + (2ac - b^2)n + ac = 0\).

For real value of n, discriminant \(\geq\) 0.

\( \Rightarrow (2ac - b^2)^2 - 4(ac)^2 \geq 0\).

\(\quad \Rightarrow \boxed{ b \geq 2\sqrt{ac}}\)

Akhil Bansal - 2 years, 11 months ago

Log in to reply

  1. First, simplify the given expression by taking L.C.M and then cross multiplication,
    Simplified expression is,
    \( \Rightarrow x^2 + x(a + b - 2c) + (ab - ac - bc) = 0 \).
    Using vieta's formula,
    Sum of roots = \( 2c - a - b = 0\).
\(\quad \quad \quad \quad \Rightarrow \color{blue}{c} = \dfrac{a + b}{2}\).

\(\quad\)Product of roots = \( ab - a\color{blue}{c} - b\color{blue}{c}\).
\( \quad \quad \quad \quad \quad \quad= ab - a\left( \dfrac{a+b}{2}\right) - b\left(\dfrac{a+b}{2}\right) \)

\( \quad \quad \quad \quad \quad \quad= -\dfrac{1}{2}(a^2 + b^2)\)

Akhil Bansal - 2 years, 11 months ago

Log in to reply

Thank u very much. :)

Abhay Kumar - 2 years, 11 months ago

Log in to reply

For the second one, you get the root as 1 by observation and then just apply Vieta's and get the answer

Harshit Singhania - 2 years, 11 months ago

Log in to reply

Can u please show it.

Abhay Kumar - 2 years, 11 months ago

Log in to reply

Sum of roots=\(2= \frac {pq-rq}{pq-rp} 》》2pq-2rp=pq-rq》》pq+rq=2rp\) Divide by pqr to get required result

Harshit Singhania - 2 years, 11 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...