 # 1)Find the sum of integral values of m for which the line y=mx+4 cuts circle x2+y2-4x=32 but not circle x2+y2=4

11 years ago

the circle x2 + y2 = 4 is in the interior of the the other circle

now crawing approx graph we can see that the y intercept of the small circle is +2or -2 however we are interested only in +2

and the big cicle has y intercept +4*2^1/2  and hence we can infer that the line y = mx + 4 is a line that rotates between the y intercepts of the two circles since the y intercept of this line is +4.

now going into the integral solutions m=0 is obviously a solun as it does not cut the small circle

an important point here is tha for all values of m the line will cut th big circle .

now putting m=1,2,3 we can see that for 1&2 by solving the two equations we see that the line intersects the small circle at no point . but for 3 it cuts th circle as D(discriminant )>0 for this case and hence for all higher values it will cut the small circle .

s our positive solutions are 1&2

now moving to the negative m by pure logic we can argue that as the small circle and the line are symmetrical about the y axis and keeping in mind the very important fact that for all values of m line cuts the big circle   we can conclude that the same nos. comes in the other side also that is the negative solutions are -1&-2.

now we have 5 solutions -2,-1,0,1,2 and the sum is clearly 0

thank you