Answer:
Explanation:
A straight line with slope m
through the point (1,3)
has equation:
y=m(x−1)+3
This line is tangent to the circle x2+y2=2
if the equation for the abscissae of intersection points has a double root. This equation is the quadratic equation:
x2+(mx−m+3)2=2⟺(m2+1)x2−2m(m−3)x+m2−6m+7=0.
It has a double root if and only if its (reduced) discriminant is 0
:
Δ′=m2(m−3)2−(m2+1)(m2−6m+7)=m2+6m−7.
1
is clearly a root of Δ′
, hence the other root is −7
. The double root of the equation in x
is then
x=m(m−3)m2+1={−175ifm=1,ifm=−7.
Finally the ordinate is calculated with the equation of the line and finally obtain as points of contact:
(−1,1)and(75,15).