Find the length of the chord made by the line x+y=4 on the circle x^2+y^2=25?

The equation of the circle is

#x^2+y^2=25#

That is, the center is #C=(0,0)# and the radius is #r=sqrt25=5#

The equation of the line is

#x+y=4#

The points of intersection of the line and the circle is obtained by solving the equations

#{(x^2+y^2=25),(x+y=4):}#

#<=>#, #{(x^2+y^2=25),(x=4-y):}#

#=>#, #(4-y)^2+y^2=25#

#16-8y+y^2+y^2-25=0#

#2y^2-8y-9=0#

The solutions to this quadratic equation in #y# are

#y=(8+-sqrt((-8)^2-4(2)(-9)))/(2*2)#

#=(8+-sqrt136)/4#

#y_1=(8+sqrt136)/4=4.915#

#=>#, #x_1=4-4.915=-0.915#

#y_2=(8-sqrt136)/4=-0.915#

#=>#, #x_2=4+0.915=4.915#

The distance between the #2# points #(-0.915, 4.915)# and #(4.915,-0.915)# is

#d=sqrt((4.915-(-0.915))^2+(-0.915-4.915)^2)#

#=8.245#

The length of the chord is #=8.245u#

graph{(x^2+y^2-25)(x+y-4)=0 [-10.43, 12.07, -5.355, 5.895]}

Let the extremities of the chord made by the line # L : x+y=4# on

the circle # S : x^2+y^2=25# be #A and B#.

Then, to find the co-ordinates of #A and B#, we have to solve

the equations of #L and S#.

From the eqn. of # L," we have, "y=4-x#.

Subst.ing in the eqn. of #S," we get, "x^2+(4-x)^2=25#.

Upon simplification, #2x^2-8x-9=0#.

The roots #x_1 and x_2# of this quadr. eqn. are the

#x"-co-ords."# of the points #A and B#.

We also have, #x_1+x_2=-(-8)/2=4, and, x_1*x_2=-9/2#.

So, if #A=A(x_1,y_1) and B=B(x_2,y_2)#, then, since #A, B in L,#

#y_1=4-x_1, and, y_2=4-x_2#, we get, by Distance Formula,

#AB^2=(x_1-x_2)^2+(y_1-y_2)^2#,

#=(x_1-x_2)^2+{(4-x_1)-(4-x_2)}^2#,

#=2(x_1-x_2)^2#,

#=2{(x_1+x_2)^2-4x_1*x_2}#,

#=2{(4)^2-4(-9/2)}#,

#=2(16+18)#,

#=68#.

# rArr AB=2sqrt17#.

To obtain the length of the chord we must calculate the intersection Points of the line #x+y=4# with the circle

#x^2+y^2=25#
To do this we plug in #y=4-x# into the equation of the circle:

#x^2+(4-x)^2=25#

expanding

#x^2+16+x^2-8x-25=0#

#2x^2-8x-9=0#

dividing by #2#

#x^2-4x-9/2=0#

using the quadatic Formula

#x_(1,2)=2pm sqrt(4+9/2)

#x_(1,2)=2pmsqrt(17/2)#

#y_1=2-sqrt(17/2)#

#y_2=2+sqrt(17/2)#

the Formula for the length is given by

#d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

plugging our coordinates we get

#d=sqrt(4*17/2+4*17/2)=2sqrt(17)#