If p x^2+2 q x+r=0 and p1 x^2+2 q1 x+r1=0 have a common root and (p1)/(p), (q1)/(q), (r1)/(r) ar

If p x^2+2 q x+r=0 and p1 x^2+2 q1 x+r1=0 have a common root...

If $p x^{2}+2 q x+r=0$ and $p_{1} x^{2}+2 q_{1} x+r_{1}=0$ have a common root and $\frac{p_{1}}{p}, \frac{q_{1}}{q}, \frac{r_{1}}{r}$ are in HP, then $p_{1}, q_{1}, r_{1}$ are
(a) in AP
(b) in GP
(c) in HP
(d) not in any progression

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. A key property of three numbers in AP is that the middle term is the average of the first and third terms. This concept is central when deducing the nature of unknown quantities based on given ratio conditions or relationships, such as those derived when combining information from a common root in quadratic equations and harmonic relationships among coefficient ratios.

Quadratic Equations and Common Roots

A quadratic equation is a second?degree polynomial equation whose general form is ax² + bx + c = 0. When two quadratics share a common root, it means there is at least one value of x that satisfies both equations. This condition leads to a relationship between the coefficients of the two quadratics and is often used to establish constraints that the coefficients must satisfy.

Harmonic Progression

A sequence of numbers is said to be in harmonic progression (HP) if their reciprocals form an arithmetic progression (AP). This property allows us to translate problems involving HP into problems about AP, which are generally easier to work with algebraically. In the context of coefficients within equations, knowing that certain ratios are in HP provides a special relation that can be exploited to determine further structure among the coefficients.

Transcript

00:01 This is not quite a question given they have a common root and this is given in hb.

00:07 We just check here if even q1r1 is in what kind of progression.

00:11 Let's check it out.

00:12 So just we use the common root another day we have.

00:16 So let's say we have alpha is a common root.

00:20 Then we can use cross multiplication.

00:22 We get alpha square over we have qr1.

00:30 Negative we have 2 q1r.

00:37 I think this is compared to with 2 qx.

00:39 We check again 2 qx.

00:44 So we have this 2 qx.

00:46 It will be 2qr1.

00:48 Next we have negative alpha over we have pr1, negative we have rp1, negative we have rp1 equals 1 over, then 2pq1, negative we have 2 q1.

01:09 So this will go now.

01:13 Now from here we call this we get alpha square equals qr1, negative q1 r1, over pq1 negative p1 q and we have alpha equals that's coming out to be r p1 p over two times pq1 negative p1 q1 negative p1 q this will go out here as a value of alpha now it is given that use this relation here p1 over p q1 over q r1 over r and in hb, harmonic progression.

02:02 So therefore, it doesn't specific protocol, p over p1, q over q1, r over r1, that'll mean ab, not an ap, so we can say q over q1, negative p over p1 equals r over r1, negative q over q1...

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