The latus rectum of the conic passing through | vedclass

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

Question

(B) The slope of the tangent at any point $(x, y)$ is $\frac{dy}{dx}$.The slope of the normal at $(x, y)$ is $-\frac{dx}{dy}$.The equation of the norm

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(B) The slope of the tangent at any point $(x, y)$ is $\frac{dy}{dx}$.The slope of the normal at $(x, y)$ is $-\frac{dx}{dy}$.The equation of the normal at $(x, y)$ is given by:$Y - y = -\frac{dx}{dy}(X - x)$Since the normal intersects the $x$-axis at $(x + 1, 0)$,we substitute $X = x + 1$ and $Y = 0$:$0 - y = -\frac{dx}{dy}(x + 1 - x)$$-y = -\frac{dx}{dy}(1)$$y = \frac{dx}{dy}$$y \, dy = dx$Integrating both sides,we get:$\int y \, dy = \int dx$$\frac{y^2}{2} = x + C$Since the conic passes through the origin $(0, 0)$,we have $0 = 0 + C$,so $C = 0$.Thus,the equation of the conic is $y^2 = 2x$.Comparing this with the standard form $y^2 = 4ax$,we get $4a = 2$,which is the length of the latus rectum.Therefore,the length of the latus rectum is $2$.

The latus rectum of the conic passing through the origin and having the property that the normal at each point $(x, y)$ intersects the $x$-axis at $(x + 1, 0)$ is:

Similar Questions

The focal chord to $y^2 = 16x$ is tangent to $(x - 6)^2 + y^2 = 2$. Then,the possible values of the slope of this chord are:

Let $l$ be the directrix of the parabola $9y^2+12y+9x-14=0$ and $l_1$ be the line passing through the vertex of this parabola and the origin. If $(h, k)$ is the point of intersection of $l$ and $l_1$,then $h+k=$

The length of the chord of the parabola $y^{2}=4ax$ $(a>0)$ which passes through the vertex and makes an acute angle $\alpha$ with the axis of the parabola is

If $x = t^2$ and $y = 2t$,then the equation of the normal at $t = 1$ is

Consider the conic $C: 25(x - 1)^2 + 25(y + 1)^2 = (3x - 4y)^2$. If the curve $E$ is the locus of the point of intersection of perpendicular tangents to the conic $C$,then the minimum distance between the curve $E$ and the point $(2, -1)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module