If p, q, r are in GP and the equations px^2+2 qx+r=0 and d x^2+2 ex+f=0 have a common root, then

step 1 I have given that PQR is in GP. These two equations have a common root show that B over P, E ove

If p, q, r are in GP and the equations px^2+2 qx+r=0 and d...

If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in $\mathrm{GP}$ and the equations $\mathrm{px}^{2}+2 \mathrm{qx}+\mathrm{r}=0$ and $\mathrm{d} \mathrm{x}^{2}+2 \mathrm{ex}+\mathrm{f}=0$ have a common root, then show that $\frac{\mathrm{d}}{\mathrm{p}}, \frac{\mathrm{e}}{\mathrm{q}}, \frac{\mathrm{f}}{\mathrm{r}}$ are in $\mathrm{AP}$

If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in $\mathrm{GP}$ and the equations $\mathrm{px}^{2}+2 \mathrm{qx}+\mathrm{r}=0$ and $\mathrm{d} \mathrm{x}^{2}+2 \mathrm{ex}+\mathrm{f}=0$ have a common root, then show that
$\frac{\mathrm{d}}{\mathrm{p}}, \frac{\mathrm{e}}{\mathrm{q}}, \frac{\mathrm{f}}{\mathrm{r}}$ are in $\mathrm{AP}$

Key Concepts

Common Root Criterion

When two quadratic equations are said to have a common root, there exists at least one value that satisfies both equations simultaneously. Analyzing the existence of a common root often involves eliminating variables or parameters, and using factorization or resultants, thereby establishing relationships among the coefficients of the equations. This method is critical in connecting seemingly unrelated sequences or conditions imposed on the parameters.

Quadratic Equations

Quadratic equations are second-degree polynomial equations that take the general form ax² + bx + c = 0, where a, b, and c are constants with a not equal to zero. The study of quadratic equations includes understanding their roots, properties such as the discriminant, and relations among coefficients and roots as given by Viète's formulas.

Arithmetic Progression

An arithmetic progression (AP) is a sequence in which the difference between any two successive members is constant. In an AP, the middle term is the arithmetic mean, or average, of its neighboring terms. This property is frequently used to set up conditions or equations that express the equality of differences between consecutive terms.

Geometric Progression

A geometric progression (GP) is a sequence in which each term after the first is found by multiplying the previous term by a constant factor called the common ratio. In many algebraic problems, properties of GP, such as the relation that the square of the middle term equals the product of the outer terms, are exploited to establish connections between corresponding parameters.

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