A constant potential difference is applied across a resistance. Consider variation of resistance with temperature. Which graph represents best the variation of power produced in resistance versus resistance?
$\left( A \right)$ $Straight$ $line$
$\left( B \right)$ $Circle$
$\left( C \right)$ $Parabola$
$\left( D \right)$ $Hyperbola$

Hint: The ratio of square of the voltage and the resistance will produce power. Using the statement determines the relation between power and resistance. Then using the relation determine the graph that represents the variation of power produced in resistance versus current. Then from the given options determine the right graph to represent the variation of power.

Complete answer:
The electrons in the metal are in random constant motion. While they are moving, they collide with each other. The potential difference is applied now across the metals and the electron tends to move towards the positive connection. During this process they are interrupted by collisions. These collisions stop their movement and this property is called resistance.
Resistance is inversely proportional to the power. Power is the product of voltage and current. Then by using ohm's law power can be defined as the ratio of square of voltage to its resistance. Resistance will increase when power is low. Resistance will decrease as power increases. The resistance of a material is connected to the current flowing through it and the potential difference between the ends.
A formula with a variable in the denominator is called a fractional formula and this corresponds to a hyperbolic relation. An inversely proportionality is always a hyperbolic relation. Power is inversely proportional to resistance. Thus, the graph is a hyperbola.

Therefore option $\left( D \right)$ is the right option.

Note: When the temperature increases in a metal the atoms will vibrate more strongly and electrons will undergo violent collisions so the resistance also increases. The ratio of square of the voltage and the resistance will produce power. An inversely proportionality is always a hyperbolic relation. Power is inversely proportional to resistance.