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Question

Tangents are drawn from A(α,0) to the parabola y2=4x and the ellipse x2a2+y2b2=1(a>b). Let the points of tangency on the ellipse be P and Q and on the parabola be R and S. If APQ and ARS are equilateral, then which of the following is (are) CORRECT?
  1. α=3
  2. Area(APQ)Area(ARS)=b436
  3. a(32,3)
  4. Distance between both of the chord of contact is 6b2


Solution

The correct options are
A α=3
B Area(APQ)Area(ARS)=b436
C a(32,3)
D Distance between both of the chord of contact is 6b2
Equation of tangent in slope form on parabola,
y=mx+1m
Putting y=0
x=1m2=α     (1)
On ellipse,
y=mx±am2+b2
As they represent same tangents,
1m2=am2+b2     (2)
We know that APQ and ARS are equilateral.
So, |m|=tan30=13
m2=13
From equation (1),
α=3

So, from equation (2),
a23+b2=3b2=3a23     (3)
We know that, a2>b2>0
Using equation (3),
a2>3a23>0a2>3a23            3a23>04a2>9                    9>a2a(32,3)(a>0)

Equation of chord of contact is 
T=0
On the parabola,
y(0)=2(x3)x=3

On the ellipse,
x(3)a2+0=1x=a23
Distance between the chord of contacts
=3(a23)=3+a23
=3+3b2     [From (3)]  
=6b2
Ratio of area of equilateral triangle = Ratio of the square of their heights
Area(APQ)Area(ARS)=(3+a23)2(33)2
=(3+3b2)2(33)2=b436     [From (3)]

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