Name: Definition and Equation of a Circle
Uploaded: 2019-07-19
Duration: 2 min 25 s
Description: Coins are circular in shape. Even many cricket stadiums are circular, and so is pizza. How are these circles made, and how can we make them perfectly circula...

Question

Find the equation of the circle concentric with the circle

2 x^{2} + 2 y^{2} + 8 x + 10 y - 35 = 0

and with area

16 π

square units.

Accepted Answer

Given $2 x^{2} + 2 y^{2} + 8 x + 10 y - 35 = 0$

Divide the equation by $2$ , we get

$x^{2} + y^{2} + 4 x + 5 y - \frac{35}{2} = 0 . . . . (1)$

On comparing with standard equation of circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$

We get, $2 g = 4, 2 f = 5 \Rightarrow g = 2, f = \frac{5}{2}$
Therefore, centre of circle is $(- g, - f) = (- 2, - \frac{5}{2})$

Let $r$ be the radius of required circle. Then area of circle $= {πr}^{2}$
also, given area of circle $= 16 π$

$\Rightarrow π r^{2} = 16 π$

$\Rightarrow r^{2} = 16$

$\Rightarrow r = 4$

Since, the required circle is concentric with the circle $(1)$ . Thus centre of required circle be $(- 2, - \frac{5}{2})$ .
Using the standard equation of circle in centre radius form ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ The equation of circle is
${(x+2)}^{2} + {(y + \frac{5}{2})}^{2} = 4^{2}$

$\Rightarrow x^{2} + 4 x + 4 + y^{2} + \frac{25}{4} + 5 y = 16$

$\Rightarrow 4 x^{2} + 4 y^{2} + 16 x + 20 y + 41 - 64 = 0$

Hence, the required equation of circle is $4 x^{2} + 4 y^{2} + 16 x + 20 y - 23 = 0$ .

Find the equation of the circle concentric with the circle $2 x^{2} + 2 y^{2} + 8 x + 10 y - 35 = 0$ and with area $16 π$ square units.

Important Questions on Circles

Learn and Prepare for any exam you want !

Find the equation of the circle concentric with the circle 2x2+2y2+8x+10y-35=0 and with area 16π square units.

Important Questions on Circles

Learn and Prepare for any exam you want !

Find the equation of the circle concentric with the circle $2 x^{2} + 2 y^{2} + 8 x + 10 y - 35 = 0$ and with area $16 π$ square units.