Step 1: Identify the Parabola and its Parametric Form
The given parabola is x2=4ay.
For this vertical parabola, any point P can be represented parametrically as (2at,at2).
The tangent at the vertex for this parabola is the line y=0 (the x-axis).
Step 2: Define the Extremities of the Focal Chord
Let the ends of the focal chord be P(t1) and Q(t2).
Coordinates are P(2at1,at12) and Q(2at2,at22).
For a focal chord of the parabola x2=4ay, the relationship between the parameters is:
Step 3: Equation of Tangents
The equation of a tangent to x2=4ay at point (2at,at2) is given by:
Step 4: Find the Abscissa on the Tangent at the Vertex
The tangent at the vertex is y=0. To find where the tangents meet this line, substitute y=0 into the tangent equation:
So, for the two tangents at t1 and t2:
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The abscissa x1=at1
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The abscissa x2=at2
Step 5: Calculate the Product x1x2
Since t1t2=−1:
Conclusion:
The product of the abscissae is −a2.
Correct Option:
(d) −a2
