Points at12- 2at1- at22-2at2- a-0 t1- t2 are collinear- if-

The correct option is D t_1t_2= 1 Points ( at _ 1 ^ 2 , 2at_1) , ( at _ 2 ^ 2 ,2at_2 ) , ( #160; a ,0 ) ( t,± t_2 ) are collinear if the area o ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard VI

Mathematics

Collinear Points

Points at12...

Question

Points ( $a t_{1}^{2}, 2 a t_{1}$ ), ( $a t_{2}^{2}, 2 a t_{2}$ ), (a,0) $t_{1} \neq t_{2}$ are collinear, if:

t_{1} t_{2} = 1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{t_{1}}{t_{2}} = - 1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

t_{1} + t_{2} = 1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

t_{1} t_{2} = - 1

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct option is D $t_{1} t_{2} = - 1$
Points $({a t}_{1}^{2}, 2 a t_{1}), ({a t}_{2}^{2}, 2 a t_{2}), (a, 0) (t, \pm t_{2})$ are collinear if the area of the triangle made by these points is zero.
The area of the triangle is
$= \frac{1}{2} | y_{1} (x_{2} - x_{3}) + y_{2} (x_{3} - x_{1}) + y_{3} (x_{1} - x_{2}) |$
$= \frac{1}{2} ∣ ∣ {2 a t}_{1} ({a t}_{2}^{2} - a) + {2 a t}_{2} (a - {a t}_{1}^{2}) + 0 ∣ ∣$
$= \frac{1}{2} ∣ ∣ {2 a^{2} t}_{1} t_{2}^{2} - {2 a^{2} t}_{1} + {2 a^{2} t}_{2} - 2 a^{2} t_{1}^{2} t_{2} ∣ ∣$
$= a^{2} | t_{1} t_{2} (t_{2} {- t}_{1}) + (t_{2} {- t}_{1}) |$
The area $= 0$
$∴ a^{2} | t_{1} t_{2} (t_{2} {- t}_{1}) + (t_{2} {- t}_{1}) | = 0$
$t_{1} t_{2} = - 1$
Hence, the answer is $t_{1} t_{2} = - 1.$

Suggest Corrections

Similar questions

If the chord joining the points $(a t_{1}^{2}, 2 a t_{1})$ and $(a t_{2}^{2}, 2 a t_{2})$ of the parabola $y^{2} = 4 a x$ passes through the focus of the parabola, then

Find the relation between $t_{1} a n d t_{2}$ if normals at $(a t_{1}^{2}, 2 a t_{1}) a n d (a t_{2}^{2}, 2 a t_{2})$ meet on the parabola.

Q. The normal a point

(b t_{1}^{2}, 2 b t_{1})

on a parabola meets the parabola again in the point

(b t_{2}^{2}, 2 b t_{2})

then

Q. If

t_{1}

and

t_{2}

are the roots of the equation

t^{2} + λ t + 1 = 0

, where

λ

is a parameter, then the line joining the points

(a t_{1}^{2}, 2 a t_{1})

and

(a t_{2}^{2}, 2 a t_{2})

always passes through

Q. Vant' Hoff equation shows the effect of temperature on equilibrium constants

K_{c}

and

K_{p} .

The

K_{p}

varies with temperature according to the relation:

Join BYJU'S Learning Program

Points (at21,2at1), (at22,2at2), (a,0) t1≠t2 are collinear, if:

Points ( $a t_{1}^{2}, 2 a t_{1}$ ), ( $a t_{2}^{2}, 2 a t_{2}$ ), (a,0) $t_{1} \neq t_{2}$ are collinear, if: