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Find the equation of the line that connects the center of the circle to (1,-3)

User Mbtamuli
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8.4k points

1 Answer

1 vote

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).

User David DV
by
8.8k points

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