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SYLLABUS
Equation of a circle with given centre and radius – General equation of circles – Centre and radius of a circle from general equation – Concentric circles – Contact of circles – Orthogonal circles – Simple problems.
Definition:
A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.
Equation of the circle with centre (h, k) and radius ‘r’ units.
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CP = r
\[\sqrt{(x\ -\ h)^2\ +\ (y\ -\ k)^2}\ =\ r\ (Using\ distance\ formula)\]
\[(x\ -\ h)^2\ +\ (y\ -\ k)^2\ =\ r^2\]
Note:
The equation of the circle with centre (0, 0 ) and radius ‘r’ units is x2 + y2 = r2
\[\color {purple} {Example\ 1\ .}\ \color {red} {What\ is\ the\ equation\ of\ the\ circle}\ with\ centre\ at\ origin\ and\ radius\ 1\ unit?\ \hspace{15cm}\]
\[\color {blue} { Soln:}\ \hspace{20cm}\]
\[We\ know\ that\ the\ equation\ of\ circle\ is\ (x\ -\ h)^2\ +\ (y\ -\ k)^2\ =\ r^2\]
\[Given\ h\ =\ 0,\ k = 0\ and\ r\ =\ 1\]
\[(x\ -\ 0)^2\ +\ (y\ -\ 0)^2\ =\ 1^2\]
\[x^2\ +\ y^2\ =\ 1\]
\[The\ equation\ of\ the\ circle\ is\ \boxed{x^2\ +\ y^2\ -\ 1=\ 0}\]
\[\color {purple} {Example\ 2\ .}\ \color {red} {Find\ the\ equation\ of\ the\ circle}\ with\ centre\ (-5, 7)\ and\ radius\ 5\ units.\ \hspace{5cm}\]
\[\color {blue} { Soln:}\ Given\ centre\ =\ (-5, 7)\ \hspace{4cm}\ and\ radius\ =\ 5\ \hspace{6cm}\]
\[We\ know\ that\ the\ equation\ of\ circle\ is\ (x\ -\ h)^2\ +\ (y\ -\ k)^2\ =\ r^2\]
\[Given\ h\ =\ -\ 5,\ k = 7\ and\ r\ =\ 5\]
\[(x\ +\ 5)^2\ +\ (y\ -\ 7)^2\ =\ 5^2\]
\[x^2\ +\ 10\ x\ +\ 25\ +\ y^2\ -\ 14y\ +\ 49\ =\ 25\ \hspace{10cm}\]
\[x^2\ +\ y^2\ +\ 10\ x\ -\ 14y\ +\ 25\ +\ 49\ -\ 25\ =\ 0\ \hspace{10cm}\]
\[x^2\ +\ y^2\ +\ 10\ x\ -\ 14y\ +\ 49\ =\ 0\ \hspace{10cm}\]
\[\therefore\
The\ required\ equation\ of\ the\ circle\ is\ \hspace{7cm}\]
\[\boxed{x^2\ +\ y^2\ +\ 10\ x\ -\ 14\ y\ +\ 49\ =\ 0}\ \hspace{5cm}\]
\[\color {purple} {Example\ 3\ .}\ \color {red} {Write\ down\ the\ equation\ of\ the\ circle}\ whose\ centre\ is\ (0, – 2)\ and\ radius\ 5.\\ \hspace{5cm}\]
\[\color {blue} { Soln:}\ Given\ centre\ =\ (0, -2)\ \hspace{4cm}\ and\ radius\ =\ 5\ \hspace{6cm}\]
\[We\ know\ that\ the\ equation\ of\ circle\ is\ (x\ -\ h)^2\ +\ (y\ -\ k)^2\ =\ r^2\]
\[Given\ h\ =\ 0,\ k = -\ 2\ and\ r\ =\ 5\]
\[(x\ -\ 0)^2\ +\ (y\ +\ 2)^2\ =\ 5^2\]
\[x^2\ +\ y^2\ +\ 4y\ +\ 4\ =\ 25\ \hspace{10cm}\]
\[x^2\ +\ y^2\ +\ 4y\ +\ 4\ -\ 25\ =\ 0\ \hspace{10cm}\]
\[x^2\ +\ y^2\ +\ 4y\ -\ 21\ =\ 0\ \hspace{10cm}\]
\[\therefore\
The\ required\ equation\ of\ the\ circle\ is\ \hspace{7cm}\]
\[\boxed{x^2\ +\ y^2\ +\ 4y\ -\ 21\ =\ 0}\ \hspace{5cm}\]
General equation of the circle: x2 + y2 + 2gx + 2fy + c = 0
Centre = (-g , -f ) and radius r = √( g2 + f2 – c)
\[\color {purple} {Example\ 4\ .}\ \color {red} {Find\ the\ centre\ and\ radius\ of\ the\ circle}\ x^2\ +\ y^2\ +\ 2\ x\ +\ 2\ y\ -\ 7\ =\ 0\ \hspace{5cm}\]
\[\color {blue} { Soln:}\ Given\ x^2\ +\ y^2\ +\ 2\ x\ +\ 2\ y\ -\ 7\ =\ 0\ \hspace{10cm}\]
\[We\ know\ that\ the\ equation\ of\ circle\ is\ x^2\ +\ y^2\ +\ 2\ g\ x\ +\ 2\ f\ y\ +\ c\ =\ 0\ \hspace{10cm}\]
\[2\ g\ =\ 2\ \hspace{3cm}\ 2\ f\ =\ 2\ \hspace{3cm}\ c\ =\ -7\]
\[g\ =\ 1\ \hspace{3cm}\ f\ =\ 1\ \hspace{3cm}\]
\[centre\ =\ (-\ g,\ -\ f)\ \hspace{4cm}\ r\ =\ \sqrt{(g^2\ +\ f^2\ -\ c)}\]
\[centre\ =\ (-\ 1,\ -\ 1)\ \hspace{4cm}\ r\ =\ \sqrt{(1^2\ +\ 1^2\ +\ 7)}\]
\[\hspace{6cm}\ r\ =\ \sqrt{(1\ +\ 1\ +\ 7)}\]
\[\hspace{6cm}\ r\ =\ \sqrt{9}\ =\ 3\]
\[\fbox{centre = (- 1, – 1) r = 3}\]
\[\color {purple} {Example\ 5\ .}\ \color {red} {Find\ the\ centre\ and\ radius\ of\ the\ circle}\ x^2\ +\ y^2\ -\ 8\ y\ +\ 3\ =\ 0\ \hspace{5cm}\]
\[\color {blue} {Soln:}\ Given\ x^2\ +\ y^2\ -\ 8\ y\ +\ 3\ =\ 0\ \hspace{10cm}\]
\[We\ know\ that\ the\ equation\ of\ circle\ is\ x^2\ +\ y^2\ +\ 2\ g\ x\ +\ 2\ f\ y\ +\ c\ =\ 0\ \hspace{10cm}\]
\[2\ g\ =\ 0\ \hspace{3cm}\ 2\ f\ =\ -\ 8\ \hspace{3cm}\ c\ =\ 3\]
\[g\ =\ 0\ \hspace{3cm}\ f\ =\ -\ 4\ \hspace{3cm}\]
\[centre\ =\ (-\ g,\ -\ f)\ \hspace{4cm}\ r\ =\ \sqrt{(g^2\ +\ f^2\ -\ c)}\]
\[centre\ =\ (0,\ 4)\ \hspace{4cm}\ r\ =\ \sqrt{(0^2\ +\ (-4)^2\ -\ 3)}\]
\[\hspace{6cm}\ r\ =\ \sqrt{(0\ +\ 16\ -\ 3)}\]
\[\hspace{6cm}\ r\ =\ \sqrt{13}\]
\[centre = (0, 4)\ \hspace{5cm}\ r\ =\ \sqrt{13}\]
Contact of Circles:
Case ( i ) :
Two circles touch externally if the distance between their centres is equal to sum of their radii.
i.e C1C2 = r1 + r2
Case ( ii ) :
Two circles touch internally if the distance between their centres is equal to difference of their radii.
i.e C1C2 = r1 – r2 or r2 – r1
\[\color {purple} {Example\ 6\ .}\ \color {red} {Prove\ that}\ the\ circles\ x^2\ +\ y^2\ -\ 4x\ -\ 6y\ +\ 9\ = 0\ \hspace{5cm}\]\[ and\ x^2\ +\ y^2\ +\ 2x\ +\ 2y\ -\ 7\ = 0\ touch\ each\ other.\ \hspace{5cm}\]
\[\color {blue} {Soln:}\ \hspace {19cm}\]
\[Given\ x^2 + y^2 -\ 4x\ -\ 6y\ +\ 9\ = 0 ——————— (1)\]
\[Given\ x^2\ +\ y^2\ +\ 2x\ +\ 2y\ -\ 7\ = 0 ——————— (2)\]
\[From\ (1)\ \hspace 10cm\]
\[2g_1 =\ -\ 4\ \hspace 2cm\ 2f_1\ =\ -\ 6\ \hspace 2cm\ c_1 =\ 9\]
\[g_1 =\ -\ 2\ \hspace 2cm\ f_1 =\ -\ 3\ \hspace 2cm\ c_1 =\ 9\]
\[Centre\ is\ C_1 = (-g_1,\ -f_1)\ \hspace 10cm\ r_1 = \sqrt{g_1^2 + f_1^2 -c_1}\]
\[ C_1 = (2,\ \ 3)\ \hspace 10cm\ r_1 = \sqrt{(-2)^2\ +\ (-3)^2\ -\ 9}\]
\[ \hspace 10cm\ r_1 = \sqrt{4\ +\ 9\ -\ 9}\]
\[ \hspace 10cm\ r_1 = \sqrt{4}\ =\ 2\]
\[\boxed{C_1 = ( 2, 3)\ and\ r_1 =\ 2}\]
\[From\ (2)\ \hspace 10cm\]
\[2g_2 =\ 2\ \hspace 2cm\ 2f_2\ =\ 2\ \hspace 2cm\ c_2 =\ -\ 7\]
\[g_2 =\ 1\ \hspace 2cm\ f_2 =\ 1\ \hspace 2cm\ c_2 =\ -\ 7\]
\[Centre\ is\ C_2 = (-g_2,\ -f_2)\ \hspace 10cm\ r_2 = \sqrt{g_2^2 + f_2^2 -c_2}\]
\[ C_2 = (-\ 1\ , -\ 1)\ \hspace 10cm\ r_2 = \sqrt{(1)^2\ +\ (1)^2\ +\ 7}\]
\[ \hspace 10cm\ r_2\ = \sqrt{1\ + 1\ +\ 7}\]
\[ \hspace 10cm\ r_2 = \sqrt{9} =\ 3\]
\[\boxed{C_2 = (-\ 1, -\ 1)\ and\ r_2 =\ 3}\]
\[C_1C_2 = \sqrt{(-1\ -\ 2)^2\ +\ (-1\ -\ 3)^2}\]
\[C_1C_2 = \sqrt{(-3)^2\ +\ (-4)^2}\]
\[C_1C_2 = \sqrt{9\ +\ 16}\]
\[C_1C_2 = \sqrt{25}\ =\ 5\]
\[r_1\ +\ r_2\ =\ 2\ +\ 3\ =\ 5\ =\ C_1C_2\]
\[\boxed{C_1C_2 = r_1 + r_2}\]
\[The\ given\ circles\ touch\ each\ other\ externally\]
\[\color {purple} {Example\ 7\ .}\ \color {red} {Prove\ that\ the\ circles}\ x^2\ +\ y^2\ -\ 10\ x\ -\ 24\ y\ +\ 120\ =\ 0\ and\ \hspace{7cm}\]\[x^2\ +\ y^2\ =\ 400\ \color {red} {touch\ each\ other}\ \hspace{5cm}\]
\[\color {blue} {Soln:}\ \hspace{20cm}\]
\[Given\ x^2\ +\ y^2\ -\ 10\ x\ -\ 24\ y\ +\ 120\ =\ 0 ——————— (1)\]
\[Given\ x^2\ +\ y^2\ =\ 400 ——————— (2)\]
\[From\ (1)\ \hspace 10cm\]
\[2g_1 = – 10\ \hspace 2cm\ 2f_1 = – 24\ \hspace 2cm\ c_1 = 120\]
\[g_1 =\ – 5\ \hspace 2cm\ f_1 =\ – 12\ \hspace 2cm\ c_1 = 120\]
\[Centre\ is\ C_1 = (-g_1,\ -f_1)\ \hspace 10cm\ r_1 = \sqrt{g_1^2 + f_1^2 -c_1}\]
\[ C_1 = (5,\ 12)\ \hspace 10cm\ r_1 = \sqrt{(\ -\ 5)^2 + (-\ 12)^2\ -\ 120}\]
\[ \hspace 10cm\ r_1 = \sqrt{25\ +\ 144\ -\ 120}\]
\[ \hspace 10cm\ r_1 = \sqrt{169\ -\ 120}\]
\[ \hspace 10cm\ r_1 = \sqrt{49} =\ 7\]
\[\boxed{C_1 = ( 5, 12)\ and\ r_1 = 7}\]
\[From\ (2)\ \hspace 10cm\]
\[2g_2 = 0\ \hspace 2cm\ 2f_2 = 0\ \hspace 2cm\ c_2 = – 400\]
\[g_2 =\ 0 \hspace 2cm\ f_2 = 0\ \hspace 2cm\ c_2 = – 400\]
\[Centre\ is\ C_2 = (-g_2,\ -f_2)\ \hspace 10cm\ r_2 = \sqrt{g_2^2 + f_2^2 -c_2}\]
\[ C_2 = (0,\ 0)\ \hspace 10cm\ r_2 = \sqrt{(0)^2 +\ (0)^2 + 400}\]
\[ \hspace 10cm\ r_2 = \sqrt{0\ +\ 0\ +\ 400}\]
\[ \hspace 10cm\ r_2\ =\ \sqrt{400}\ =\ 20\]
\[\boxed{C_2 = ( 0, 0)\ and\ r_2\ =\ 20}\]
\[C_1C_2 = \sqrt{(0\ -\ 5)^2 + (0\ -\ 12)^2}\]
\[C_1C_2 = \sqrt{(-\ 5)^2 + (-\ 12)^2}\]
\[C_1C_2 = \sqrt{25\ +\ 144}\]
\[C_1C_2 = \sqrt{169}\ =\ 13\]
\[r_2 – r_1 \ =\ 20\ -\ 13\ =\ 7\ = C_1C_2\]
\[\boxed{C_1C_2 = r_2 – r_1}\]
\[The\ given\ circles\ touch\ each\ other\ internally\]
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