If the line y−√3x+3=0 cuts the parabola y2=x+2 at A and B, the PA⋅PB is equal to [If P=√3,0)].
A
4(2−√3)3
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B
4(√3+2)3
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C
4√33
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D
2(√3+2)3
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Solution
The correct option is B4(√3+2)3 Given, P=(√3,0) Equation of line AB is x−√3cos60∘=y−0sin60∘=r(say) ⇒x=√3+r2,yr√32 Therefore, point (√3+r2,r√32) lies on y2=x+2 ⇒3r24=√3+r2+2 ⇒3r24−r2−(2+√3)=0 Let the roots be r1 and r2, then the product r1×r2=PA⋅PB=∣∣
∣
∣
∣∣−(2+√3)34∣∣
∣
∣
∣∣ =4(2+√3)3.