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.

$(1, - 6), (5, 2), (7, 0), (- 1, - 4)$ .

Accepted Answer

Let the given points be, $(1, - 6), (5, 2), (7, 0), (- 1, - 4)$

Three non-collinear points are always concyclic as they form the triangle and circle passing through those points is circumcircle and the centre of that circumcircle is circumcentre.

Let $O \equiv (x, y)$ be the circumcentre.

Considering $∆ A B C$ and circumcentre $O (x, y)$ .

Using radius condition, $O A = O B = O C$

$\Rightarrow O A^{2} = O B^{2}$

$\Rightarrow (x - 1)^{2} + (y + 6)^{2} = (x - 5)^{2} + (y - 2)^{2}$

$\Rightarrow x^{2} + 1 - 2 x + y^{2} + 36 + 12 y = x^{2} + 25 - 10 x + y^{2} + 4 - 4 y$

$\Rightarrow 37 - 29 = 2 x - 12 y - 10 x - 4 y$

$\Rightarrow x + 2 y = - 1$ ...........(i)

Now, $O B^{2} = O C^{2}$

$\Rightarrow (x - 5)^{2} + (y - 2)^{2} = (x - 7)^{2} + (y - 0)^{2}$

$\Rightarrow x^{2} + 25 - 10 x + y^{2} + 4 - 4 y = x^{2} + 49 - 14 x + y^{2}$

$\Rightarrow 4 x - 4 y + 29 - 49 = 0$

$\Rightarrow x - y = 5$ ..........(ii)

Equations, $(i) - (i i)$

$\Rightarrow (x + 2 y) - (x - y) = - 1 - 5$

$\Rightarrow 3 y = - 6$

$\Rightarrow y = - 2$

From (i), $x + 2 (- 2) = - 1$

$\Rightarrow x - 4 = - 1$

$\Rightarrow x = 3$

Therefore, centre $= (3, - 2)$

Radius of circle

$\Rightarrow O A^{2} = (3 - 1)^{2} + (- 2 + 6)^{2}$

$\Rightarrow O A^{2} = 20$

$\Rightarrow$ Radius of the circle $= O A = \sqrt{20}$

Using equation of circle in centre-radius form, we get the equation of circle as

$(x - 3)^{2} + (y + 2)^{2} = 20$

Verifying whether $4^{t h}$ point is on the circle or not,

$\Rightarrow (- 1 - 3)^{2} + (- 4 + 2)^{2} = 20$

$\Rightarrow {(- 4)}^{2} + {(- 2)}^{2} = 20$

$\Rightarrow 16 + 4 = 20$

$∴ 4^{t h}$ point lies on the circumcircle of the triangle formed by the three points taken.

$∴$ Given points are concyclic.

And, the equation of the circle on which they lie is $(x - 3)^{2} + (y + 2)^{2} = 20$ .

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

$(1, - 6), (5, 2), (7, 0), (- 1, - 4)$ .

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,-6),(5,2),(7,0),(-1,-4).

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.

$(1, - 6), (5, 2), (7, 0), (- 1, - 4)$ .