
Prove that the equation of the director circle to the parabola is x + a = 0.
Answer
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Hint: The director circle of a parabola is the directrix of the curve. First draw the point of intersection of the tangent to the parabola whose equation is . Consider the end points of the focal part on the parabola as and . Start with differentiating the equation of parabola.
Complete step-by-step answer:
and .
Note: The chord joining and passes through focus, if . This is formed because the tangents are perpendicular.
Complete step-by-step answer:
We know that the equation of parabola is

The figure of the parabola given below illustrates the focus, directrix, axis and vertex of the parabola.
We know that the tangents at the extremities of any focal chord intersect at right angles on the directrix. Let us take the end points of the focal part on the parabola as and .
To find the equation of tangent to parabola at a given point, we need to differentiate the equation of parabola,
Differentiating both sides,
Therefore, the slope of a tangent at a given point on the parabola is:
Equation of line passing through having slope m by point – slope form
Therefore, equation of tangent
At point , the equation of tangent is
At point , the equation of tangent is
Therefore, the slope
Now, let three tangents intersect at a point .
The coordinates of points of intersection of tangents in a parabola can be found by subtracting the equation of tangents at the 2 points P and P’ respectively.
i.e. Equation (ii) – Equation (iii)
Subtracting both the equation
Substitute the value of y in equation (ii).
Putting the value of x to find y in equation (ii).
(x, y) can be taken as (h, k)
Since, the tangents are perpendicular,
Where h = x.
Hence, the locus of the point (h, k) is x = -a which is the equation of the directrix.
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