Question

The solution of \[\dfrac{{dy}}{{dx}}{\text{ }} = \dfrac{{\left[ {ax + h} \right]}}{{\left[ {by + k} \right]}}\]represents a circle when
1. \[{\mathbf{a}}{\text{ }} = {\text{ }}{\mathbf{b}}\]
2. \[{\mathbf{a}}{\text{ }} = {\text{ }}-{\text{ }}{\mathbf{b}}{\text{ }} \ne {\text{ }}{\mathbf{0}}\]
3. \[{\mathbf{a}}{\text{ }} = {\text{ }}-{\text{ }}{\mathbf{2b}}{\text{ }} = {\text{ }}{\mathbf{0}}\]
4. \[\;{\mathbf{a}}{\text{ }} = {\text{ }}{\mathbf{2b}}\]

Answer 1

Hint: Distance from any point on the circumference of the circle to the center of the circle is called the radius of the circle. When drawn a triangle inside the circle with the radius as the two sides of the circle and coordinates of center as (a,b) and point on the circumference having (h.k) as coordinates leads to an equation using Pythagoras theorem \[\sqrt {{{(h - a)}^2} + {{(k - b)}^2}} = r\]
Squaring both sides of the equation give
\[{(h - a)^2} + {(k - b)^2} = {r^2}\]
This equation represents a circle. There are many other ways to represent a circle. The general form of the equation of the circle can be written as \[{x^2}{\text{ }} + {\text{ }}{y^2}{\text{ }} + {\text{ }}2gx{\text{ }} + {\text{ }}2fy{\text{ }} + {\text{ }}c{\text{ }} = {\text{ }}0\] where x and y are the arbitrary points present on the circumference and g, f, and c are the constants.

Complete step by step solution:
Step 1: Given equation of circle is
\[\dfrac{{dy}}{{dx}}{\text{ }} = \dfrac{{\left[ {ax + h} \right]}}{{\left[ {by + k} \right]}}\]
Cross multiply the both sides as shown below
\[\left[ {by + k} \right]dy{\text{ }} = \left[ {ax + h} \right]dx\]
Step 2: integrate both sides using the sum rule of integration.
\[\int {\left[ {by + k} \right]dy} {\text{ }} = \int {\left[ {ax + h} \right]dx} \]
Integrating with respect to y on left hand side and with respect to x on right hand side of the equation
$\dfrac{{b{y^2}}}{2} + ky + c = \dfrac{{a{x^2}}}{2} + hx + d$ {c and d are constants}
$\dfrac{{b{y^2}}}{2} - \dfrac{{a{x^2}}}{2} = hx - ky + C${C is a constant combining c and d}
This implies \[{\mathbf{a}}{\text{ }} = {\text{ }}-{\text{ }}{\mathbf{b}}{\text{ }} \ne {\text{ }}{\mathbf{0}}\]
Hence the correct answer of the question is option \[\left( {\mathbf{2}} \right)\].

Note: the integration must be taken care of to arrive at the correct answer as follow\[\smallint {x^n}\;dx{\text{ }} = {\text{ }}\dfrac{{{x^{n + 1}}}}{{n + 1}}{\text{ }} + {\text{ }}C\]. The constant C has significance when the upper and lower limits are not specified over the integral. Integration of a constant is the same as the constant. This means integration has no effect on a constant value. \[\smallint cf\left( x \right){\text{ }}dx{\text{ }} = {\text{ }}c\smallint f\left( x \right){\text{ }}dx\]