Name: Understanding Parametric Form of a Circle
Uploaded: 2019-07-19
Duration: 6 min 4 s
Description: This video deals with the concept of parametric equations of circles, how they are written, how we can write the parametric form of a circle from cartesian f...

Question

Find the equation of the circle which circumscribes the triangle formed by the line

\frac{x}{a} + \frac{y}{b} = 1

and the coordinate axes. Hence, find its centre and radius.

Accepted Answer

Given sides of triangle $A B C$ are $\frac{x}{a} + \frac{y}{b} = 1 . . . . (1)$

$x = 0 . . . . (2)$

$y = 0 . . . . (3)$

Equations $(1)$ and $(2)$ intersects at $B (0, b)$

Equations $(2)$ and $(3)$ intersects at $O (0, 0)$ and equations $(1)$ and $(3)$ intersects at $A (a, 0)$ .

Let the required equation of circle be $x^{2} + y^{2} + 2 g x + 2 f y + c = 0 . . . . (4)$

Since the circle given by equation $(4)$ passes through the points $O (0, 0); A (a, 0)$ and $B (0, b)$ .

$∴ c = 0;$ $a^{2} + 2 g a = 0$

$\Rightarrow g = - \frac{a}{2}$

and $b^{2} + 2 f = 0$

$\Rightarrow f = - \frac{b^{2}}{2}$

Putting the values of $g, f$ and $c$ in equation $(4),$ we get

$x^{2} + y^{2} - a x - b y = 0$ which is the required equation of circle.

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

Centre of circle is $(- g, - f)$ and $r =$ radius of circle $= \sqrt{g^{2} + f^{2} - c}$

$∴$ Centre of circle is $(\frac{a}{2}, \frac{b}{2})$ and radius of circle $= \sqrt{\frac{a^{2}}{4} + \frac{b^{2}}{4}}$

$= \frac{1}{2} \sqrt{a^{2} + b^{2}}$

Find the equation of the circle which circumscribes the triangle formed by the line $\frac{x}{a} + \frac{y}{b} = 1$ and the coordinate axes. Hence, find its centre and radius.

Important Questions on Circles

Learn and Prepare for any exam you want !

Find the equation of the circle which circumscribes the triangle formed by the line xa+yb=1 and the coordinate axes. Hence, find its centre and radius.

Important Questions on Circles

Learn and Prepare for any exam you want !

Find the equation of the circle which circumscribes the triangle formed by the line $\frac{x}{a} + \frac{y}{b} = 1$ and the coordinate axes. Hence, find its centre and radius.