If 4 ℓ 2 − 5 m 2 + 6 ℓ + 1 = 0 , then ℓ x + m y + 1 = 0 is tangent to a fixed circle. If tangents from P varying on x + y − 1 = 0 touch the fixed circle at A , B , find the fixed point on the contact

Question

If 4 ℓ 2 − 5 m 2 + 6 ℓ + 1 = 0 , then ℓ x + m y + 1 = 0 is tangent to a fixed circle. If tangents from P varying on x + y − 1 = 0 touch the fixed circle at A , B , find the fixed point on the contact chord A B .

TestHub · Accepted Answer

The circle is x 2 + y 2 − 6 x + 4 = 0 . Its chord of contact from P = ( u , 1 − u ) has equation ux + ( 1 − u ) y − 3 ( x + u ) + 4 = 0 , equivalently u ( x − y − 3 ) − 3 x + y + 4 = 0 . The fixed point solves x − y = 3 and − 3 x + y + 4 = 0 , giving 2 x = 1 , y = − 5/2 , hence (1/2, -5/2).

Mathematics - Circle Question with Solution | TestHub

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