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Question

A normal is drawn to the parabola y2=9x at the point P(4,6). A circle is described on SP as a diameter, where S is the focus. If the length of the intercept(in units) made by the circle on the normal at point P is L units, then the value of 8L is (units)

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Solution

Eq. of Normal to given parabola is
(y6)=62×94(x4)
3y18=164x
3y+4x34=0
Focus of parabola is (94,0)
Equation of circle with SP as diameter is
(x4)(x94)+(y6)y=0
x2+y26yx(4+94)+9=0
x2+y2254x6y+9=0
(x258)2+(y3)2=(258)2

Clearly centre is (258,3) and radius is 258units
Distance of centre from line 3y+4x34=0 is
d=∣ ∣ ∣252+9345∣ ∣ ∣=2510=52units
Length of intercept =2r2d2=2258258254=225811625=25435L=154 units
8L=30 units

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