locus of the point of intersection of the perpendicular tangents to the circles x^2+y^2=a^2,x^2+y^2=b^2 - Maths - Conic Sections

Question

locus of the point of intersection of the perpendicular tangents to
the circles x^2+y^2=a^2,x^2+y^2=b^2

Rahul Raj · Accepted Answer

dear student

for first circle, let tangent bey=mx+a1+m2then tangent for circle will bey=-xm+b1+1m2 as the two tangents are perpendicularsolving for point of intersection we getmy=m2x+am1+m2my=-x+b1+m2solving we gety=bm-a1+m2and x=b-am1+m2we have to elimate m to get locusbm-ay=b-amxm=ax+byay+bxsoy=bm-a1+m2=bax+byay+bx-a1+ax+byay+bx2simplifying we get(ax+by)2+(ay+bx)2=(b2-a2)2

regards