  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?α=−3Area(△APQ)Area(△ARS)=b436a∈(32,3)Distance between both of the chord of contact is 6−b2

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 6−b2Equation 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∘=1√3 ⇒m2=13 From equation (1), α=−3 So, from equation (2), a23+b2=3⇒b2=3−a23     ⋯(3) We know that, a2>b2>0 Using equation (3), a2>3−a23>0⇒a2>3−a23            3−a23>0⇒4a2>9                    9>a2⇒a∈(32,3)(∵a>0) Equation of chord of contact is  T=0 On the parabola, y(0)=2(x−3)⇒x=3 On the ellipse, x(−3)a2+0=1⇒x=−a23 Distance between the chord of contacts =∣∣∣3−(−a23)∣∣∣=3+a23 =3+3−b2     [From (3)]   =6−b2 Ratio of area of equilateral triangle = Ratio of the square of their heights Area(△APQ)Area(△ARS)=(−3+a23)2(−3−3)2 =(−3+3−b2)2(−3−3)2=b436     [From (3)]  Suggest corrections   