Give for parabola $y^{2} = 4 a x$ ,One of the tangents from the point (l,m)

touch the parabola at $(a {t_{1}}^{2}, 2 a {t_{1}}^{2}) .$ The other point of contact in

terms of $a, t_{1}, l, m$ is $(\frac{l^{2}}{a {t_{1}}^{2}}, \frac{2 l}{t_{1}})$ .

True

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False

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Solution

The correct option is A
True

It is useful to known that the point of intersection of tangents

from $(a {t_{1}}^{2}, 2 a {t_{1}}^{2}) & (a {t_{2}}^{2}, 2 a {t_{2}}^{2}) i s (a t_{1} t_{2}, a (t_{1} + t_{2})) .$

Given ,point of intersection as $, (l, m) .$

$\Rightarrow a t_{1} t_{2} = l \Rightarrow t_{2} = \frac{l}{a t_{1}}$

$∴$ The point of contact of second tangent to the parabola is

$(a {t_{2}}^{2}, 2 a {t_{2}}^{2}) = (\frac{l^{2}}{a {t_{1}}^{2}}, \frac{2 l}{t_{1}})$ .

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Similar questions

Q. If two tangents from any point on the directrix to the parabola

y^{2} = 4 a x

touch the parabola at

t_{1}, t_{2}

then

Q. Intersection points of tangents and normal at the extremities of latus rectum of the parabola

y^{2} = 4 a x

will form a square.

Q. If the chord of contact of tangents from a point

P

to the parabola

y^{2} = 4 a x

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x^{2} = 4 b y,

then the locus of

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A pair of tangents are drawn from a point on the directrix to a parabola $y^{2} = 4 a x$ . The angle formed by the tangents will always be $90^{\circ}$

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Give for parabola y2=4ax,One of the tangents from the point (l,m) touch the parabola at (at12,2at12). The other point of contact in terms of a,t1,l,m is (l2at12,2lt1).

The correct option is A True It is useful to known that the point of intersection of tangents from (at12,2at12) & (at22,2at22) is (at1t2,a(t1+t2)). Given ,point of intersection as,(l,m). ⇒ at1t2=l⇒ t2=lat1 ∴ The point of contact of second tangent to the parabola is (at22,2at22)=(l2at12,2lt1).

Give for parabola $y^{2} = 4 a x$ ,One of the tangents from the point (l,m)

touch the parabola at $(a {t_{1}}^{2}, 2 a {t_{1}}^{2}) .$ The other point of contact in

terms of $a, t_{1}, l, m$ is $(\frac{l^{2}}{a {t_{1}}^{2}}, \frac{2 l}{t_{1}})$ .