The number of points, where the locus of the point of intersection of two tangents to a parabola $y^{2} = 4 x$ which intercept a constant length on any fixed tangent cuts the y-axis will be

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Solution

Let $y t = x + a t^{2}$ ......(1)
be a fixed tangent to the parabola $y^{2} = 4 a x$ ...... (2)
Let the other two tangent to (2) be
$y t_{1} = x + {a t_{1}}^{2}$ .......(3)
And $y t_{2} = x + {a t_{2}}^{2}$ ...... (4)
The point of intersection of (1) with (3) and (4) are
$P {a t t_{1}, a (t + t_{1})}$ and $Q {a t t_{2}, a (t + t_{2})}$
Given PQ is constant or $P Q^{2} = c o n s t a n t$
$\Rightarrow a^{2} t^{2} (t_{1} - t_{2})^{2} + a^{2} (t_{1} - t_{2})^{2} = c o n s t a n t$
$\Rightarrow a^{2} (t^{2} + 1) (t_{1} - t_{2})^{2} = c o n s t a n t$
$\Rightarrow (t_{1} - t_{2})^{2} = c o n s t a n t$ since t is constant as (1) is a fixed tangent.
$\Rightarrow (t_{1} - t_{2})^{2} = c (s a y)$ ...... (5)
Let $(x_{1}, y_{1})$ be the point of intersection of (3) and (4),
then $x_{1} = a t_{1} t_{2} a n d y_{1} = a (t_{1} + t_{2})$ ...... (6)
We know that
$(t_{2} - t_{1})^{2} = (t_{1} + t_{2})^{2} - 4 t_{1} t_{2}$ ...... (7)
From equation (5), (6) and (7) we get
$c = (y_{1} / a)^{2} - 4 x_{1} / a$
or ${y_{1}}^{2} = 4 x_{1} a + a^{2} c = 4 a (x_{1} + 1 / 4 a c)$
$\Rightarrow t h e l o c u s o f (x_{1}, y_{1})$ is
$y^{2} = 4 a (x + 1 / 4 a c)$
Now c > 0 and for x = 0
$y^{2} = c$
so two points

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