The equation of the parabola having focus at (−1,−2) and the directrix x−2y+3=0 is ax2+bxy+cy2+gx+fy+16=0. If α and β are the roots of the equation gx2−fx+60=0, then the value of α3+β3 is
A
2(4ab−5c)
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B
3(4ab−5c)
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C
3(5ab−4c)
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D
2(5ab−4c)
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Solution
The correct option is D2(5ab−4c) Given, Focus ≡(−1,−2)
and Directrix is x−2y+3=0
By definition of the parabola, we know √(x+1)2+(y+2)2=∣∣∣x−2y+3√5∣∣∣ ⇒5(x2+2x+1+y2+4y+4)=(x−2y+3)2 ⇒4x2+4xy+y2+4x+32y+16=0
So, we have a=4,b=4,c=1,g=4,f=32
Now, α,β are the roots of 4x2−32x+60=0 ⇒x2−8x+15=0 ⇒α=3,β=5 or α=5,β=3 α3+β3=27+125=152 2(5ab−4c)=2(80−4)=152