What is the condition for the points (a, 0),( | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Points are collinear if the area of the triangle formed by them is zero.Area $= \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| = 0

Vedclass · Accepted Answer

$t_1t_2 = -1$

Vedclass · Answer

(C) Points are collinear if the area of the triangle formed by them is zero.Area $= \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| = 0$.Substituting the points $(a, 0)$,$(at_1^2, 2at_1)$,and $(at_2^2, 2at_2)$:$\frac{1}{2} |a(2at_1 - 2at_2) + at_1^2(2at_2 - 0) + at_2^2(0 - 2at_1)| = 0$.Dividing by $2a^2$ (assuming $a 
eq 0$):$|(t_1 - t_2) + t_1^2 t_2 - t_2^2 t_1| = 0$.$t_1 - t_2 + t_1 t_2 (t_1 - t_2) = 0$.$(t_1 - t_2)(1 + t_1 t_2) = 0$.Since $t_1 
eq t_2$ for distinct points,we must have $1 + t_1 t_2 = 0$,which implies $t_1 t_2 = -1$.

What is the condition for the points $(a, 0)$,$(at_1^2, 2at_1)$,and $(at_2^2, 2at_2)$ to be collinear?

Similar Questions

What is the locus of the vertices $(x, y)$ of the parabolas $y = \frac{a^3x^2}{3} + \frac{a^2x}{2} - 2a$?

The length of the latus rectum of the conic $25[(x-2)^2+(y-3)^2]=(3x-4y+7)^2$ is

The focal distance of the point $(4, 4)$ on the parabola with vertex at $(0, 0)$ and symmetric about the $y$-axis is:

Let the equation of the tangent at a point $P$ on the parabola $x^2-4x-4y+16=0$ be $2x-y-5=0$. If the equation of the normal drawn at $P$ to this parabola is $ax+y+c=0$,then find the value of $ac$.

For the parabola $y^2+6y-2x=-5$,consider the following statements:
$I$. The vertex is $(-2, -3)$.
$II$. The directrix is $y+3=0$.
Which of the following is correct?

Vedclass Test Series

Exam Paper Generator

Online Exam Module