Cool Parabola
A normal is drawn to the parabola at the point. . A circle is described on as diameter; where is the focus. The length of the intercept made by the circle on the normal at point can be expressed in the form of where and are co-prime. Find .
.
The answer is 19.
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1 solution
write the equation of the normal at P on parabola. that will come out as 4x+3y=34 find the centre of the circle. that will come out as (25/8,3) then write the Foot of the perpendicular on the normal by the formula (H-(aE)/(a^2+b^2) , K-(bE)/(a^2+b^2)) (H,K) is the coordinate of point P and E is aH+bK+c=0 where ax+by+c=0 is the equation of line on which perpendicular lie {This formula comes from image formula of a point which can be extended into 3-D as well} the foot of perpendicular comes out to be (41/8,9/2) then just take this point^s distance from (4,6) and double it You will get the required result.