Let P(4,43) be a point on the parabola y2=4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:
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Step-by-Step Solution
Step 1: Determine the parabola's parameter 'a' and coordinates of P
The given point P(4,43) lies on the parabola y2=4ax. We can substitute the coordinates of P into the parabola's equation to find the value of the parameter a. This gives us a=3. Thus, the equation of the parabola is y2=12x. The focus is F(a,0)=F(3,0) and the directrix is x=−a, which is x=−3.
Step 2: Find the coordinates of point Q using focal chord property
For a parabola y2=4ax, any point can be represented parametrically as (at2,2at). We found a=3, so P(3t12,6t1). Using the coordinates of P, we find t1=323. Since PQ is a focal chord, the product of the parameters t1 and t2 is −1. This allows us to find t2=−23. Substituting t2 back into the parametric form gives the coordinates of Q as (49,−33).
Step 3: Determine the coordinates of M and N on the directrix
The directrix of the parabola is the line x=−3. Since M and N are the feet of the perpendiculars from P and Q to the directrix, their x -coordinates will be −3, and their y -coordinates will be the same as P and Q respectively. So, M(−3,43) and N(−3,−33).
Step 4: Calculate the area of the quadrilateral PQMN
The quadrilateral PQMN is a trapezium because PM and QN are perpendicular to the directrix, making them parallel to each other. The area of a trapezium is given by 21(sum of parallel sides)×(height). Here, PM and QN are the parallel sides, and MN is the height. We calculate the lengths of PM, QN, and MN and substitute them into the formula to find the area.