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

The axis of a parabola is along the line $y=x$. The distance of its vertex $A$ from $(0,0)$ is $\sqrt{2}$ and that of its focus $S$ from $(0,0)$ is $2\sqrt{2}$. If $A$ and $S$ lie in the first quadrant,then the equation of the parabola in parametric form is

Find the point of intersection of the parabola $y^2 = 2px$ and a circle whose center is at the focus of the parabola and which touches the directrix of the parabola.

The length of the latus rectum of the conic $25[(x-2)^2+(y-3)^2]=(3x-4y+7)^2$ is

If the tangent to the curve $x = at^2, y = 2at$ is perpendicular to the $x$-axis,then what is the point of contact?

$A$ parabola having its axis parallel to the $Y$-axis passes through the points $(0, 2/5)$,$(4, -2)$,and $(1, 8/5)$. Which of the following points lies on this parabola?

Vedclass Test Series

Exam Paper Generator

Online Exam Module