If the tangents at the two points x=p and x=q to the curve...

If the tangents at the two points $x=p$ and $x=q$ to the curve $y=(x-p)(x-q)(x-r)$ are parallel to one another then (a) $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in $\mathrm{GP}$ (b) $\mathrm{q}, \mathrm{p}, \mathrm{r}$ are in $\mathrm{AP}$ (c) $\mathrm{r}, \mathrm{q}, \mathrm{p}$ are in $\mathrm{AP}$ (d) $\mathrm{p}, \mathrm{r}, \mathrm{q}$ are in $\mathrm{AP}$

If the tangents at the two points $x=p$ and $x=q$ to the curve $y=(x-p)(x-q)(x-r)$ are parallel to one another then
(a) $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in $\mathrm{GP}$
(b) $\mathrm{q}, \mathrm{p}, \mathrm{r}$ are in $\mathrm{AP}$
(c) $\mathrm{r}, \mathrm{q}, \mathrm{p}$ are in $\mathrm{AP}$
(d) $\mathrm{p}, \mathrm{r}, \mathrm{q}$ are in $\mathrm{AP}$

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Key Concepts

Geometric Progression (GP)

A geometric progression is a sequence of numbers where each term is obtained by multiplying the previous term by a constant factor. Although not always the case, checking if parameters form a GP can sometimes be necessary when derivative conditions lead to multiplicative relationships among the terms.

Derivative

Differentiation is the process of finding the instantaneous rate of change of a function, which is used to determine the slope of the tangent line at any given point on a curve. In problems involving tangents, the derivative is crucial for computing and equating slopes at different points.

Parallel Tangents

Tangents to a curve are parallel if they have the same slope. This condition allows one to set the derivatives at the specified points equal to each other, forming an equation that can reveal relationships among the parameters of the function.

Arithmetic Progression (AP)

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. In the context of relating the parameters of a function, establishing that certain parameters form an AP implies a linear relationship among them, which may arise from conditions obtained by equating derivatives.

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