Prove that the locus of a point which moves so that its distance from a fixed line is equal to - Maths - Conic Sections

Question

Prove that the locus of a point which moves so that its distance
from a fixed line is equal to the length of the tangent drawn from
it to a given circle is a parabola Find the position of the focus
and directrix

Shruti Tyagi · Accepted Answer

Dear Student,

Let AB be the fixed line, and O be the cenre of fixed circle
Taking O asOrigin, aline through O and parallel to AB as axis of y and a line through O perpendicular to AB as axis, the equation of circle may be written asx2+y2=r2  --1and equation of line as x=a --2where radius of circle is r and distance of centre from line AB is a
Let P be any point h,k on the locus, PT be the tangent from P to 1 and PN be perpendicular from P upon AB
 ThenPT=h2+k2-r2 and PN=h-aNow by hypothesis PT=PN or PT2=PN2Simplifying and generalising , we geth2+k2-r2=h-a2⇒h2+k2-r2=h2+a2-2ahnow put h=x and k=yy2-r2=a2-2axy2=-2ax+a2+r2 --3now this is the equation of a parabola, hence proved that locus of of point above is parabola
equation 3 can be written as y2=-2ax-a2+r22athe focus is a2+r22a-2a4,0=a2+r22a-a2,0=r22a,0And the equation of directrix x=a2+r22a+a2
Regards,

Prove that the locus of a point which moves so that its distance from a fixed line is equal to the length of the tangent drawn from it to a given circle is a parabola . Find the position of the focus and directrix