Name: How to Find Locus of Any Point
Uploaded: 2019-07-19
Duration: 13 min 14 s
Description: In this video, we will learn about the locus of a point in 3D, which will help you to solve questions in exams based on this topic. Watch and learn.

Question

Using distance formula show that

(3, 3)

is the centre of the circle passing through the points

(6, 2), (0, 4)

and

(4, 6)

. Find the radius ( in units ) of the circle.

Accepted Answer

Given that circle passes through the points $A (6, 2), B (0, 4), C (4, 6)$ .

Let us assume $O (x, y)$ be the centre of the circle.

We know that distance from the centre to any point on the circle is equal.

So, $OA = OB = OC$

We know that distance between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is $\sqrt{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}$

Now,

$\Rightarrow OA = OB$

$\Rightarrow {OA}^{2} = {OB}^{2}$

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

$\Rightarrow x^{2} - 12 x + 36 + y^{2} - 4 y + 4 = x^{2} + y^{2} - 8 y + 16$

$\Rightarrow 12 x - 4 y = 24$

$\Rightarrow 3 x - y = 6$ .......(1)

Now,

$\Rightarrow OB = OC$

$\Rightarrow {OB}^{2} = {OC}^{2}$

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

$\Rightarrow x^{2} + y^{2} - 8 y + 16 = x^{2} - 8 x + 16 + y^{2} - 12 y + 36$

$\Rightarrow 8 x + 4 y = 36$

$\Rightarrow 2 x + y = 9$ ...... .(2)

On solving (1) and (2), we get

$\Rightarrow x = 3$ and $y = 3$

$∴ (3, 3)$ is the centre of the circle.

We know radius is the distance between the centre and any point on the circle.

Let ‘r’ be the radius of the circle.

$\Rightarrow r = OA = \sqrt{(3 - 6)^{2} + (3 - 2)^{2}}$

$\Rightarrow r = \sqrt{(- 3)^{2} + (1)^{2}}$

$\Rightarrow r = \sqrt{9 + 1}$

$\Rightarrow r = \sqrt{10}$ units.

$∴$ The radius of the circle is $\sqrt{10}$ units.

Using distance formula show that $(3, 3)$ is the centre of the circle passing through the points $(6, 2), (0, 4)$ and $(4, 6)$ . Find the radius ( in units ) of the circle.

Important Questions on Cartesian System of Rectangular Coordinates

Learn and Prepare for any exam you want !

Using distance formula show that 3, 3 is the centre of the circle passing through the points 6,2,0,4 and 4,6. Find the radius ( in units ) of the circle.

Important Questions on Cartesian System of Rectangular Coordinates

Learn and Prepare for any exam you want !

Using distance formula show that $(3, 3)$ is the centre of the circle passing through the points $(6, 2), (0, 4)$ and $(4, 6)$ . Find the radius ( in units ) of the circle.