Conics

[Aladdin]

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Given the two parabolas : (P) : y^2 = 2x - 1 & (P') : x^2 = 2y - 1

2) Verify that the point A(1,1) is common to (P) and (P'),and prove that (OA)is a common tangent to these two parabolas.

3)Prove that the line (d),perpendicular to (OA) at O is a common tangent to (P) and (P').


My Attempt is :

For #2) I know how to verify the point A ... But the rest !

Any help will be greatly appreciated .
Thank You !

 

[BigGlenntheHeavy]

\(\displaystyle Here \ is \ the \ graph, \ Aladdin, \ can \ you \ take \ it \ from \ here?\)

[attachment=0:21bw85t1]mno.jpg[/attachment:21bw85t1]

 

[Aladdin]

Thanks for the graph Glenn,this makes me sure of my work . . .

On the graph I can see the two lines of equation y=x and (d):y=-x respectively ,Right Glenn .

So,to prove that (OA)is a common tangent to these two parabolas, I must find the equation of the line and replace it in the parabolas to see that it's true . . .

 

[Aladdin]

#5) After sketching (d) , (P) and (P').

The area of the region bounded by (P),the axis of abscissas and the line of equation x=1 is equal to 3cm^2,
Deduce the are,in cm^2 of the region bounded by (P),(P'),the axis of abscissas and the axis of ordinates.

I think there's an error for the deduce part !

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