Ex – 1 Draw the graphs of two circles and determine whether they touch each other externally or internally

\[\text{To draw the graph of given equation of the circles}\ \hspace{15cm}\\ x^2\ +\ y^2\ -\ 4x\ +\ 6y\ +\ 8\ =\ 0\ and\ \hspace{10cm}\\ x^2\ +\ y^2\ -\ 4x\ +\ 6y\ +\ 8\ =\ 0\ \hspace{10cm}\]

\[a)\ \text{To graph the equations of the circles in the Cartesian plane.}\ \hspace{15cm}\]

\[b)\ \text{To determine the coordinates of the centres and radii of the circles and}\ \hspace{15cm}\\ \text{mark them on the graph}\ \hspace{10cm}\]

\[c)\ \text{To determine the distance between the centres of the circles}\ \hspace{15cm}\]

\[d)\ \text{To determine whether the circles are touching eah other or not}\ \hspace{15cm}\]

\[e)\ \text{If the circles are touching each other, determine whether they are}\ \hspace{15cm}\\ \text{touching internally or externally.}\ \hspace{10cm}\]

\[f)\ \text{Verify whether any of the relationships}\ \hspace{20cm}\\ C_1C_2\ =\ r_1\ +\ r_2\ or\ C_1C_2\ =\ |r_1\ -\ r_2|\ holds\ or\ not.\ \hspace{5cm}\]

\[\color{green}{Step\ 1:}\ \text{Open Geogebra classic 5 (by double clicking on the icon)}\ \hspace{15cm}\]

\[\color{green}{Step\ 2:}\ \text{To draw the graph of the given circle}\ x^2\ +\ y^2\ -\ 4x\ +\ 6y\ +\ 8\ =\ 0\ \hspace{18cm}\\ \text{by using the input bar type x^2 + y^2 – 4x + 6y + 8 = 0 and press Enter Key.}\ \hspace{10cm}\\ \text{The graph of the given circle appears in the graphics view and}\ \hspace{10cm}\\ \text{named as eq1 in the algebra view}\ \hspace{10cm}\]

\[\color{green}{Step\ 3:}\ \text{Similarly, to draw the graph of the given circle}\ x^2\ +\ y^2\ -\ 10x\ -\ 6y\ +\ 14\ =\ 0\ \hspace{18cm}\\ \text{by using the input bar type x^2 + y^2 – 10x – 6y + 14 = 0 and press Enter Key.}\ \hspace{10cm}\\ \text{The graph of the given circle appears in the graphics view and}\ \hspace{10cm}\\ \text{named as eq2 in the algebra view}\ \hspace{10cm}\]

\[\color{green}{Step\ 4:}\ \text{To determine the center of the circle eq1 by using the input bar type}\ \hspace{18cm}\\ C1\ =\ Center(eq1)\ \text{and press Enter Key.}\ \hspace{15cm}\\ \text{Centre of the circle C1}\ \text{appears in the graphics view}\ \hspace{10cm}\\ \text{and coordinates of the C1 appears in the algebra view}\ \hspace{10cm}\]

\[\color{green}{Step\ 5:}\ \text{Similarly, to determine the center of the circle eq2 by using the input bar type}\ \hspace{18cm}\\ C2\ =\ Center(eq2)\ \text{and press Enter Key.}\ \hspace{15cm}\\ \text{Centre of the circle C2 appears in the graphics view}\ \hspace{10cm}\\ \text{and coordinates of the C 2 appears in the algebra view}\ \hspace{10cm}\]

\[\color{green}{Step\ 6:}\ \text{To measure the radius of the circle eq1 by using the input bar type}\ \hspace{18cm}\\ R1\ =\ radius(eq1)\ \text{and press Enter Key.}\ \hspace{15cm}\\ \text{value of the radius of the circle eq 1 appears in the algebra view}\ \hspace{10cm}\]

\[\color{green}{Step\ 7:}\ \text{Similarly, to measure the radius of the circle eq2 by using the input bar type}\ \hspace{18cm}\\ R2\ =\ radius(eq2)\ \text{and press Enter Key.}\ \hspace{15cm}\\ \text{value of the radius of the circle eq 2 appears in the algebra view}\ \hspace{10cm}\]

\[\color{green}{Step\ 8:}\ \text{To draw the radius of the circle 1 and circle 2,}\ \hspace{16cm}\\ \text{choose point tool, select point on the object tool.}\ \hspace{10cm}\]

\[\color{green}{Step\ 9:}\ \text{Plot the points on the two circles and}\ \hspace{18cm}\\ \text{name it as A and B appears on the graphics view.}\ \hspace{10cm}\]

\[\color{green}{Step\ 10:}\ \text{To draw a segment point A to center C1,}\ \hspace{18cm}\\ \text{choose, line tool, select segment tool.}\ \hspace{10cm}\]

\[\color{green}{Step\ 11:}\ \text{Select the points A and C1 successively,}\ \hspace{18cm}\\ \text{the segment appears in the graphics view and renamed as r1}\ \hspace{10cm}\]

\[\color{green}{Step\ 12:}\ \text{Similarly select the points B and C2 successively,}\ \hspace{18cm}\\ \text{the segment appears in the graphics view and renamed as r2}\ \hspace{10cm}\]

\[\color{green}{Step\ 13:}\ \text{Similarly, select the points C1 and C2 successively,}\ \hspace{18cm}\\ \text{the segment appears in the graphics view and renamed as d,}\ \hspace{15cm}\\ \text{the value of the d appears in the algebra view.}\ \hspace{10cm}\]

\[\color{green}{Step\ 14:}\ \text{Verify whether}\ d = R1\ +\ R2\ or\ d = |R1\ -\ R2|\ \hspace{19cm}\\ \text{by using the input bar type}\ R1\ +\ R2\ or\ abs(R1\ -\ R2)\ \text{and press Enter Key.}\ \hspace{10cm}\\ \text{the values of the} R1\ +\ R2\ or\ abs(R1\ -\ R2)\ \hspace{10cm}\\ \text{appears in the algebra view.}\ \hspace{10cm}\]

\[\color{green}{Step\ 15:}\ \text{To create input box for circle}\ C1\ and\ with\ eq1,\ \hspace{18cm}\\ \text{choose the action object tool then select the input box tool}\ \hspace{10cm}\]

\[\color{green}{Step\ 16:}\ \text{The popup window appears on the graphics view,}\ \hspace{18cm}\\ \text{select input Box, click on the graphics view}\ \hspace{15cm}\\ \text{the window appears. Type the caption Circle 1, then linked Object eq1.}\ \hspace{10cm}\\ \text{Then press Enter key, input box appears on the graphics view}\ \hspace{10cm}\]

\[\color{green}{Step\ 17:}\ \text{Similarly create an input box for Circle 2}\ \hspace{18cm}\]