Let y2=12x be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP)(SQ)=4147. Let C be the circle described taking PQ as a diameter. If the equation of a circle C is 64x2+64y2−αx−643y=β, then β−α is equal to:
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Step-by-Step Solution
Step 1: Identify Parabola Properties and Focal Chord Length
First, we identify the properties of the given parabola y2=12x. Comparing it with the standard form y2=4ax, we find that 4a=12, which means a=3. The focus S of the parabola is at (a,0), so S=(3,0). For a focal chord PQ with endpoints P(at12,2at1) and Q(at22,2at2), the lengths SP and SQ are given by a(1+t12) and a(1+t22) respectively. Since PQ is a focal chord, the product of the parameters t1t2=−1.
Step 2: Calculate Product of Focal Distances
We are given that (SP)(SQ)=4147. We substitute the expressions for SP and SQ and use the relation t2=−1/t1. This simplifies the product to 9(1+t12)(1+1/t12). Expanding this, we get 9(2+t12+1/t12).
Step 3: Solve for t12
Now we equate the calculated product (SP)(SQ) to the given value 4147 and solve for t12. This leads to a quadratic equation in t12. Factoring the quadratic equation 12(t12)2−25t12+12=0, we find two possible values for t12: 43 or 34. Let's choose t12=43.
Step 4: Determine Coordinates of P and Q
Using t12=43, we get t1=23. Then t2=−t11=−32. Now we can find the coordinates of points P and Q using the parametric form (at2,2at). Substituting a=3 and the values of t1 and t2, we get P(49,33) and Q(4,−43).
Step 5: Find the Equation of Circle C
The circle C is described taking PQ as a diameter. The equation of a circle with endpoints of a diameter (x1,y1) and (x2,y2) is given by (x−x1)(x−x2)+(y−y1)(y−y2)=0. Substituting the coordinates of P and Q, we expand and simplify the equation. Multiplying by 64 to match the given form, we get 64x2+64y2−400x+643y−1728=0.
Step 6: Compare and Calculate β−α
Comparing our derived equation 64x2+64y2−400x+643y−1728=0 with the given equation 64x2+64y2−αx−643y=β, we can identify the values of α and β. We find that α=400 and β=1728. Finally, we calculate β−α=1728−400=1328.