The tangent at P to a parabola $y^{2} = 4 a x$ meets the directrix at U and the latus rectum at V then SUV (where is is the focus) :

must be a right triangle

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must be an equilateral friangle

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must be an isosceles triangle

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must be a right isosceles triangle.

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Solution

The correct option is B must be an isosceles triangle
The tangent at P to a parabola $y^{2} = 4 a x$ meets the directrix at U and latus rectm at V then SUV :
Equation of tangent at $(a t^{2}, 2 a t)$ to parabola $y^{2} = 4 a x$
$y t = x + a t^{2}$
At directrix $x = - a$
$y = \frac{a t^{2} - a}{t}$
At latus rectm $x = a$
$y = \frac{a t^{2} + a}{t}$
$S \equiv (a, 0) U \equiv (- a, \frac{a t^{2} - a}{t}) V \equiv (a, \frac{a t^{2} + a}{t}) S U = \sqrt{(2 a)^{2} + {(\frac{a t^{2} - a}{t})}^{2}} = a (\frac{t^{2} + 1}{t}) S V = a (\frac{t^{2} + 1}{t}) U V = 2 a \sqrt{\frac{t^{2} + 1}{t}} S U = S V \neq U V$
Therefore, SUV must be isosceles triangle.

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The tangent at P to a parabola y2=4ax meets the directrix at U and the latus rectum at V then SUV (where is is the focus) :

The tangent at P to a parabola $y^{2} = 4 a x$ meets the directrix at U and the latus rectum at V then SUV (where is is the focus) :