If the line y = sqrt{3}x cuts the curve x^4 + | vedclass

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

Question

(C) Any point on the line $y = \sqrt{3}x$ can be represented in polar form as $(r \cos 	heta, r \sin 	heta)$,where $	heta = 60^\circ = \frac{\pi}{3

Vedclass · Accepted Answer

$96$

Vedclass · Answer

(C) Any point on the line $y = \sqrt{3}x$ can be represented in polar form as $(r \cos 	heta, r \sin 	heta)$,where $	heta = 60^\circ = \frac{\pi}{3}$.Thus,the coordinates are $\left(r \cos 60^\circ, r \sin 60^\circight) = \left(\frac{r}{2}, \frac{r\sqrt{3}}{2}ight)$.Substituting these into the curve equation $x^4 + ax^2y + bxy + cx + dy + 6 = 0$:$\left(\frac{r}{2}ight)^4 + a\left(\frac{r}{2}ight)^2\left(\frac{r\sqrt{3}}{2}ight) + b\left(\frac{r}{2}ight)\left(\frac{r\sqrt{3}}{2}ight) + c\left(\frac{r}{2}ight) + d\left(\frac{r\sqrt{3}}{2}ight) + 6 = 0$$\frac{r^4}{16} + \frac{a\sqrt{3}r^3}{8} + \frac{b\sqrt{3}r^2}{4} + \frac{r(c + d\sqrt{3})}{2} + 6 = 0$Multiplying by $16$:$r^4 + (2a\sqrt{3})r^3 + (4b\sqrt{3})r^2 + 8(c + d\sqrt{3})r + 96 = 0$This is a quartic equation in $r$ where the roots $r_1, r_2, r_3, r_4$ represent the distances $OA, OB, OC, OD$ from the origin.By Vieta's formulas,the product of the roots is the constant term divided by the leading coefficient:$OA \cdot OB \cdot OC \cdot OD = r_1 r_2 r_3 r_4 = \frac{96}{1} = 96$.

If the line $y = \sqrt{3}x$ cuts the curve $x^4 + ax^2y + bxy + cx + dy + 6 = 0$ at $A$,$B$,$C$,and $D$,then the value of $OA \cdot OB \cdot OC \cdot OD$ is,(where $O$ is the origin).

Similar Questions

The slopes of the common tangents to the parabola $(x - 1)^2 = 4(y - 2)$ and the ellipse $\frac{(x - 1)^2}{1} + \frac{(y - 2)^2}{2} = 1$ are $m_1$ and $m_2$. Then,$m_1^2 + m_2^2$ is equal to:

The graph of the conic $x^2-(y-1)^2=1$ has one tangent line with positive slope that passes through the origin. If the point of tangency is $(a, b)$,then find the value of $\sin^{-1}\left(\frac{a}{b}\right)$.

The equation of the common tangent to the curves $y^2 = 8x$ and $xy = -1$ is:

Let $L$ be a tangent line to the parabola $y^{2}=4x-20$ at the point $(6,2)$. If $L$ is also a tangent to the ellipse $\frac{x^{2}}{2}+\frac{y^{2}}{b}=1$,then the value of $b$ is equal to ..... .

If a tangent to the hyperbola $xy = -1$ is also a tangent to the parabola $y^2 = 8x$,then the equation of that tangent is

Vedclass Test Series

Exam Paper Generator

Online Exam Module