Question

Find the radius of the circle 4x2+4y2−5x−9y=8

Answer 1

Answer:

The radius of the circle is $\dfrac{\sqrt{234} }{8}$ .

Step-by-step explanation:

Equation of a circle with center $(a,b)$ and radius $r$ is

$(x-a)^2+(y-b)^2=r^2$

Expanding the equation, we get

$\implies x^2-2xa+a^2+y^2-2yb+b^2=r^2\\\\\implies x^2+y^2-2xa-2yb=r^2-a^2-b^2$

We have given the equation,

$4x^2+4y^2-5x-9y=8$

$\implies x^2+y^2-\frac{5}{4 }x-\frac{9}{4}y=\frac{8}{4}=2$

Comparing the equation with the standard form, we can see that

$-2a=\dfrac{-5}{4} \\\\\implies a=\dfrac{-5}{4}\div -2=\dfrac{-5}{4}\times \dfrac{-1}{2}=\dfrac{5}{8}$

Also,

$\implies -2b=\dfrac{-9}{4} \\\\\implies b=\dfrac{9}{4}\times \dfrac{-1}{2}=\dfrac{9}{8}$

And

$r^2-a^2-b^2=2\\\\\implies r^2=2+a^2+b^2=2+(\frac{5}{8}) ^2+(\frac{9}{8}) ^2=2+\dfrac{5^2+9^2}{8^2} \\\\=\dfrac{2\times 64+25+81}{64}= \dfrac{234}{64}$

So,

$r=\frac{\sqrt{234} }{8}$

The radius of the circle is $\dfrac{\sqrt{234} }{8}$ .

Answer 2

Given:

4x2+4y2−5x−9y=8

To Find:

The radius of the circle

Solution:

We are given a circle equation and we need to find the radius of the circle, the equation of a circle with radius r is expressed as

                         $(x-h)^2+(y-k)=r^2$

So we can convert the given equation into this form to find the radius of the circle, which goes as,

$4x^2+4y^2-5x-9y=8\\(2x)^2-2*2x*\frac{5}{4}+(\frac{5}{4})^2+(2y)^2-2*2y*\frac{9}{4} +(\frac{9}{4})^2=8+\frac{25}{16} + \frac{81}{16} \\(2x-\frac{5}{4})^2+(2y-\frac{9}{4} )^2=8+\frac{25}{16} + \frac{81}{16}$

so the value of the radius will be the right part of the equation if we compare the standard equation, so we can express it as,

$r^2=8+\frac{25}{16} + \frac{81}{16} \\r=\sqrt{\frac{234}{16} } \\=\frac{\sqrt{234} }{4}$

Hence, the radius of the given circle is root234/4.