Let r be the radius of the circle, which touches x -axis at point ( a , 0), a 0 and the parabola y ^2=9 x at the point (4,6) . Then r is equal to ________

Step-by-step solution

aligned & (x-a)^2+(y-r)^2=r^2 & (4-a)^2+(6-r)^2=r^2 & 16+a^2-8 a+36+r^2-12 r=r^2 & a^2-8 a-12 r+52=0 aligned Tangent to parabola at (4,6) is 6.4=9 . ( x+4 2 ) i.e. 3 x-4 y+12=0 This is also tangent to the circle aligned & C P=r & 3 a-4 r+12 5 = r aligned 3 a +12=4 r 5 r array l ar - r array ......(1) equation of circle is (x-a)^2+(y-r)^2=r^2 satsty P (4,6) a ^2-8 a -12 r+52=0 (2) From equation (1) If a +4=3 r then a =+6 (rejected) If 3 a+12=-r then a=-14 and r=30