The chord PQ of the parabola y^2 = x,where on | vedclass

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

Question

(A) The equation of the parabola is $y^2 = x$.Given that the chord $PQ$ is perpendicular to the axis of the parabola (the $x$-axis) and one end $P$ is

Vedclass · Accepted Answer

$-4$

Vedclass · Answer

(A) The equation of the parabola is $y^2 = x$.Given that the chord $PQ$ is perpendicular to the axis of the parabola (the $x$-axis) and one end $P$ is $(4, -2)$,the $x$-coordinate of $Q$ must also be $4$.Since $Q$ lies on the parabola $y^2 = x$,substituting $x = 4$ gives $y^2 = 4$,so $y = \pm 2$. Since $P$ is $(4, -2)$,$Q$ must be $(4, 2)$.The derivative of the parabola $y^2 = x$ with respect to $x$ is $2y \frac{dy}{dx} = 1$,which gives $\frac{dy}{dx} = \frac{1}{2y}$.The slope of the tangent at $Q(4, 2)$ is $\frac{dy}{dx} \Big|_{(4, 2)} = \frac{1}{2(2)} = \frac{1}{4}$.The slope of the normal at $Q$ is the negative reciprocal of the slope of the tangent.Therefore,the slope of the normal is $-\frac{1}{1/4} = -4$.

The chord $PQ$ of the parabola $y^2 = x$,where one end $P$ of the chord is at point $(4, -2)$,is perpendicular to the axis of the parabola. Then the slope of the normal at $Q$ is

Similar Questions

The length of the normal chord of the parabola $y^2 = 4x$,which makes an angle of $\frac{\pi}{4}$ with the $x$-axis,is:

Statement $I$: $4x^2+y^2-4xy-30x-50y+40=0$ is the equation of a parabola having $(2,3)$ as its focus and $x+2y+5=0$ as its directrix.
Statement $II$: The equation of the directrix of the parabola $x^2-4x+16y+52=0$ is $y+1=0$.
Which of the above statements is (are) true?

If $2x + y + \lambda = 0$ is a normal to the parabola $y^{2} = 8x$,then the value of $\lambda$ is

If the line $2x + y + k = 0$ is normal to the parabola $y^2 = -8x$,then the value of $k$ will be

Two parabolas have the same focus. If their directrices are the $x$-axis and the $y$-axis respectively,then the slope of their common chord is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module