# Find the equation and comman chord of a circle

• May 3rd 2009, 10:31 PM
zorro
Find the equation and comman chord of a circle
Find the equation and length of common chord of the circle
$\displaystyle 2x^2 + 2y^2 + 7x -5y + 2 = 0$ and $\displaystyle x^2 + y^2 - 4x + 8y - 18 =0$
• May 5th 2009, 10:37 AM
VonNemo19
First, solve for y in each of the equations. Start by moving all x terms and constants to the right side

$\displaystyle 2y^2-5y=-2x^2-7x-2$

complete the square
divide by the coefficient of the quadratic term

$\displaystyle (y-\frac{5}{4})^2=-x^2-\frac{7}{2}x-1+\frac{25}{16}$

taking the root and adding

$\displaystyle y=\pm{\sqrt{-2x^2-\frac{7}{2}x-1+\frac{25}{16}}}+\frac{5}{4}$

Do the same with the other equation and solve the system.
• May 5th 2009, 12:45 PM
Opalg
Quote:

Originally Posted by zorro
Find the equation and length of common chord of the circle
$\displaystyle 2x^2 + 2y^2 + 7x -5y + 2 = 0$ and $\displaystyle x^2 + y^2 - 4x + 8y - 18 =0$

Something wrong here. Those two circles don't intersect at all, so how can they have a common chord? (The centres of the circles are at (2,–4) and $\displaystyle \bigl(-\tfrac74,\tfrac54\bigr)$. The distance between these points is about 6.86. This is a lot more than the sum of the two radii, which are $\displaystyle \sqrt2$ and $\displaystyle \sqrt{58}/4$, both of them less than 2.)

Edit. Oops! First radius is $\displaystyle \sqrt{38}$, not $\displaystyle \sqrt2$. Thanks to skeeter for spotting that.
• May 5th 2009, 03:18 PM
skeeter
uhh ... the two circles do intersect.
• May 5th 2009, 03:26 PM
skeeter
Quote:

Originally Posted by zorro
Find the equation and length of common chord of the circle
$\displaystyle 2x^2 + 2y^2 + 7x -5y + 2 = 0$ and $\displaystyle x^2 + y^2 - 4x + 8y - 18 =0$

$\displaystyle 2x^2 + 2y^2 + 7x -5y + 2 = 0$

$\displaystyle -2(x^2 + y^2 - 4x + 8y - 18 =0)$

---------------------------------

$\displaystyle 15x - 21y + 38 = 0$

linear equation is the equation of the chord
• Dec 27th 2009, 11:51 PM
zorro
what about the length of the chord ????
• Dec 28th 2009, 03:24 AM
HallsofIvy
Quote:

Originally Posted by zorro
what about the length of the chord ????

You know the equation of the chord and so can find the points where that chord intersects either of the circles (it intersects both circles in the same points, of course). Find the distance between those two points.