The parabolas ax^2 + 2bx + cy = 0 and dx^2 + | vedclass

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

Question

(C) Given the parabolas $ax^2 + 2bx + cy = 0$ and $dx^2 + 2ex + fy = 0$ intersect on the line $y = 1$.At $y = 1$,the equations become $ax^2 + 2bx + c

Vedclass · Accepted Answer

$\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in $A.P.$

Vedclass · Answer

(C) Given the parabolas $ax^2 + 2bx + cy = 0$ and $dx^2 + 2ex + fy = 0$ intersect on the line $y = 1$.At $y = 1$,the equations become $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$.Since $a, b, c$ are in $G.P.$,we have $b^2 = ac$,so $b = \sqrt{ac}$.The first equation becomes $ax^2 + 2\sqrt{ac}x + c = 0$,which is $(\sqrt{a}x + \sqrt{c})^2 = 0$.Thus,$x = -\sqrt{\frac{c}{a}}$.Substituting this value of $x$ into the second equation $dx^2 + 2ex + f = 0$:$d(\frac{c}{a}) + 2e(-\sqrt{\frac{c}{a}}) + f = 0$.Dividing by $c$,we get $\frac{d}{a} + \frac{f}{c} = 2e\frac{1}{\sqrt{ac}}$.Since $b = \sqrt{ac}$,this simplifies to $\frac{d}{a} + \frac{f}{c} = \frac{2e}{b}$.This condition implies that $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in $A.P.$

The parabolas $ax^2 + 2bx + cy = 0$ and $dx^2 + 2ex + fy = 0$ intersect on the line $y = 1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in $G.P.$,then

Similar Questions

For the parabola $y^2 = 8(x - 3)$,let $P$ be a point on it. Let $M$ be the foot of the perpendicular from $P$ to the directrix,and $S$ be the focus of the parabola. If $\triangle SPM$ is an equilateral triangle,find the length of each side of the triangle.

If $L$ is the normal drawn to the parabola $y^2 = 8x$ at the point $t = \frac{1}{\sqrt{2}}$,then the foot of the perpendicular drawn from the focus of the parabola on to the normal $L$ is

The equation of the normal to the parabola $x^2 = 8y$ at $x = 4$ is

What is the equation of the pair of tangents drawn from the point $(1, 4)$ to the parabola $y^2 = 12x$?

$A$ circle of radius $4$,drawn on a chord of the parabola $y^2 = 8x$ as diameter,touches the axis of the parabola. Then,the slope of the chord is

Vedclass Test Series

Exam Paper Generator

Online Exam Module