The equation #x^2+y^2=25# defines a circle at the origin and radius of 5. The line #y = x + 1# passes through the circle. What are the point(s) at which the line intersects the circle?

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Answer 1 · 2015-10-24T20:02:06

There are 2 points of intrersection: #A=(-4;-3)# and #B=(3;4)#

To find if there are any points of intersection you have to solve system of equations including circle and line equations:

#{(x^2+y^2=25),(y=x+1):}#

If you substitute #x+1# for #y# in first equation you get:

#x^2+(x+1)^2=25#

#x^2+x^2+2x+1=25#

#2x^2+2x-24=0#

You can now divide both sides by #2#

#x^2+x-12=0#

#Delta=1^2-4*1*(-12)#

#Delta=1+48=49#

#sqrt(Delta)=7#

#x_1=(-1-7)/2=-4#

#x_2=(-1+7)/2=3#

Now we have to substitute calculated values of #x# to find corresponding values of #y#

#y_1=x_1+1=-4+1=-3#

#y_2=x_2+1=3+1=4#

Answer: There are 2 points of intersect: #(-4;-3)# and #(3;4)#