if the chord of contact of tangents from a point P(h,k) to the circle x2 + y2 = a2 touches - Maths - Conic Sections

Question

if the chord of contact of tangents from a point P(h,k)
to the circle x2 + y2 = a2 touches
the circle x2 + (y-a)2 = a2, then
locus of P is

Muskaan Talwar · Accepted Answer

Dear Student,

The equation of chord of contact from P(h,k) to circle x2+y2=a2 is xh+yk=a2
⇒x=a2-ykhThe chord of contact touches the circle x2+y-a2=a2⇒a2-ykh2+y-a2=a2⇒a4+y2k2-2yka2+h2y2+h2a2-2h2ay=a2h2⇒a4+y2k2-2yka2+h2y2-2h2ay=0⇒y2k2+h2-2ah2+a2ky+a4=0which is quadratic in y
Since the line touches the circle at only one point, so it must have equal roots
⇒Discriminant=0⇒-2ah2+a2k2-4k2+h2a4=0⇒4a2h4+4a4k2+8a3h2k-4k2a4-4h2a4=0⇒4a2h4+8a3h2k-4h2a4=0⇒4h2+8ak-4a2=0⇒h2+2ak-a2=0⇒h2+2ak=a2Thus locus of P isx2+2ay=a2

Hope this information will clear your doubts about topic

Regards

if the chord of contact of tangents from a point P(h,k) to the circle x2 + y2 = a2 touches the circle x2 + (y-a)2 = a2, then locus of P is

if the chord of contact of tangents from a point P(h,k) to the circle x² + y² = a² touches the circle x² + (y-a)² = a², then locus of P is