Normals at P, Q, R are drawn to the parabola | vedclass

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

Question

(B) The equation of a normal to the parabola $y^2 = 4ax$ (where $a=1$) is $y = mx - 2am - am^3$. Substituting $a=1$ and the point $(3, 0)$,we get $0 =

Vedclass · Accepted Answer

obtuse angled triangle

Vedclass · Answer

(B) The equation of a normal to the parabola $y^2 = 4ax$ (where $a=1$) is $y = mx - 2am - am^3$. Substituting $a=1$ and the point $(3, 0)$,we get $0 = 3m - 2m - m^3$. $m^3 - m = 0 \Rightarrow m(m-1)(m+1) = 0$. Thus,the slopes of the normals are $m = 0, 1, -1$. The coordinates of the feet of the normals are given by $(am^2, -2am)$. For $m=0$,$P = (0, 0)$. For $m=1$,$Q = (1, -2)$. For $m=-1$,$R = (1, 2)$. Calculating the side lengths: $PQ = \sqrt{(1-0)^2 + (-2-0)^2} = \sqrt{1+4} = \sqrt{5}$. $PR = \sqrt{(1-0)^2 + (2-0)^2} = \sqrt{1+4} = \sqrt{5}$. $QR = \sqrt{(1-1)^2 + (2 - (-2))^2} = \sqrt{0+16} = 4$. Since $(QR)^2 = 16$ and $(PQ)^2 + (PR)^2 = 5 + 5 = 10$,we have $(QR)^2 > (PQ)^2 + (PR)^2$. Therefore,$\Delta PQR$ is an obtuse angled triangle.

Normals at $P, Q, R$ are drawn to the parabola $y^2 = 4x$ which intersect at the point $(3, 0)$. Then,$\Delta PQR$ is:

Similar Questions

If $PQ$ is a focal chord of the parabola $y^2=4x$ with focus $S$ and $P=(4,4)$,then $SQ=$

Let $P(\alpha, \beta)$ be a point on the parabola $y^2 = 4x$. If $P$ also lies on the chord of the parabola $x^2 = 8y$ whose midpoint is $(1, 5/4)$,then $(\alpha - 28)(\beta - 8)$ is equal to:

What is the area of the triangle formed by the vertex and the endpoints of the latus rectum of the parabola $x^2 = 8y$?

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

$A$ circle has its center $C$ on the axis of a parabola and it touches the parabola at point $P$. The line segment $CP$ makes an angle of $120^{\circ}$ with the axis of the parabola. If the radius of the circle is $2$,then the latus rectum of the parabola is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module