Name: Two Circles Touching Internally and Externally
Uploaded: 2019-07-19
Duration: 6 min 6 s
Description: In this video, we will learn the condition for two circles touching internally and externally using an interesting example. Come, let's learn and watch this...

Question

Show that the following four points are concyclic and find the equation of the circle on which they lie

$(1, 1), (- 6, 0), (- 2, 2), (- 2, - 8)$

Accepted Answer

Let the equation of the required circle be $(1, 1), (- 6, 0), (- 2, 2), (- 2, - 8)$

Since the circle passes through $A (1, 1)$

$\Rightarrow {(1)}^{2} + {(1)}^{2} + 2 g (1) + 2 f (1) + c = 0$

$\Rightarrow 2 + 2 g + 2 f + c = 0$

$\Rightarrow 2 g + 2 f + c = - 2 \to (2)$

Also, the circle passes through $B (- 6, 0)$

$\Rightarrow {(- 6)}^{2} + {(0)}^{2} + 2 g (- 6) + 2 f (0) + c = 0$

$\Rightarrow 36 - 12 g + c = 0$

$- 12 g + c = - 36 \to (3)$

The circle passes through $C (- 2, 2)$

$\Rightarrow {(- 2)}^{2} + {(2)}^{2} + 2 g (- 2) + 2 f (2) + c = 0$

$\Rightarrow 4 + 4 - 4 g + 4 f + c = 0$

$\Rightarrow - 4 g + 4 f + c = - 8 \to (4)$

Now, equation $(3) - equation (4)$ , gives

$- 8 g - 4 f = - 28$

$\Rightarrow 2 g + f = 7 \to (5)$

Now, equation $(2) - Equation (3)$ , gives

$14 g + 2 f = 34 \to (6)$

Solving equations $(5)$ and $(6)$ , we get

$f = 3$ and $g = 2$

Put them in equation $(2)$ , we get

$2 (2) + 2 (3) + c = - 2$

$\Rightarrow 4 + 6 + c = - 2$

$\Rightarrow c = - 12$

Hence, from $(1)$ , we have

$x^{2} + y^{2} + 2 (2) x + 2 (3) y - 12 = 0$

$\Rightarrow x^{2} + y^{2} + 4 x + 6 y - 12 = 0 \to (7)$

Now, if the points $(1, 1), (- 6, 0), (- 2, 2), (- 2, - 8)$ are concyclic. So, the point $(- 2, - 8)$ must satisfy the equation of the circle i.e, equation $(7)$

$\Rightarrow {(- 2)}^{2} + {(- 8)}^{2} + 4 (- 2) + 6 (- 8) - 12 = 4 + 64 - 8 - 48 - 12 = 68 - 68 = 0$

$∴$ The given points are concyclic and equation of the required circle is

$x^{2} + y^{2} + 4 x + 6 y - 12 = 0$ .

Show that the following four points are concyclic and find the equation of the circle on which they lie

$(1, 1), (- 6, 0), (- 2, 2), (- 2, - 8)$

Important Questions on Circle

Learn and Prepare for any exam you want !

Show that the following four points are concyclic and find the equation of the circle on which they lie (1,1),(-6,0),(-2,2),(-2,-8)

Important Questions on Circle

Learn and Prepare for any exam you want !

Show that the following four points are concyclic and find the equation of the circle on which they lie

$(1, 1), (- 6, 0), (- 2, 2), (- 2, - 8)$