Let A and B be two distinct points on the par | vedclass

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

Question

(C) Let the coordinates of points $A$ and $B$ be $(t_1^2, 2t_1)$ and $(t_2^2, 2t_2)$ respectively.The center of the circle with diameter $AB$ is the m

Vedclass · Accepted Answer

$(C, D)$

Vedclass · Answer

(C) Let the coordinates of points $A$ and $B$ be $(t_1^2, 2t_1)$ and $(t_2^2, 2t_2)$ respectively.The center of the circle with diameter $AB$ is the midpoint of $AB$,which is $\left(\frac{t_1^2+t_2^2}{2}, t_1+t_2\right)$.The radius of the circle is $r$. Since the axis of the parabola (the $x$-axis,$y=0$) touches the circle,the absolute value of the $y$-coordinate of the center must be equal to the radius $r$.Thus,$|t_1+t_2| = r$,which implies $t_1+t_2 = \pm r$.The slope $m$ of the line joining $A$ and $B$ is given by $m = \frac{2t_2 - 2t_1}{t_2^2 - t_1^2} = \frac{2(t_2 - t_1)}{(t_2 - t_1)(t_2 + t_1)} = \frac{2}{t_1+t_2}$.Substituting $t_1+t_2 = \pm r$,we get $m = \pm \frac{2}{r}$.Therefore,the possible values for the slope are $\frac{2}{r}$ and $-\frac{2}{r}$,which correspond to options $C$ and $D$.

Let $A$ and $B$ be two distinct points on the parabola $y^2 = 4x$. If the axis of the parabola touches a circle of radius $r$ having $AB$ as its diameter,then the slope of the line joining $A$ and $B$ can be

Similar Questions

Consider the parabola with vertex $\left(\frac{1}{2}, \frac{3}{4}\right)$ and the directrix $y=\frac{1}{2}$. Let $P$ be the point where the parabola meets the line $x=-\frac{1}{2}$. If the normal to the parabola at $P$ intersects the parabola again at the point $Q$,then $(PQ)^{2}$ is equal to :

If the line $x + y = k$ is a normal to the parabola $y^2 = 4x$,find the value of $k$.

What is the slope of the tangents drawn from $(3, 8)$ to the parabola $y^2 = -12x$?

$A$ point on the parabola whose focus and vertex are respectively at $\left(\frac{5}{4}, -2\right)$ and $(1, -2)$ is

The equation of the normal to the parabola $y^2 = 4ax$ at the point $\left( \frac{a}{m^2}, \frac{2a}{m} \right)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module