If the circle x^2 + y^2 = a^2 intersects the | vedclass

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

Question

(D) Given the equations $x^2 + y^2 = a^2$ and $xy = c^2$.Substituting $y = \frac{c^2}{x}$ into the circle equation:$x^2 + \frac{c^4}{x^2} = a^2$$x^4 -

Vedclass · Accepted Answer

All of the above

Vedclass · Answer

(D) Given the equations $x^2 + y^2 = a^2$ and $xy = c^2$.Substituting $y = \frac{c^2}{x}$ into the circle equation:$x^2 + \frac{c^4}{x^2} = a^2$$x^4 - a^2 x^2 + c^4 = 0$This is a biquadratic equation in $x$. The roots are $x_1, x_2, x_3, x_4$.From the properties of roots,the sum of roots $x_1 + x_2 + x_3 + x_4 = 0$ (coefficient of $x^3$ is $0$) and the product of roots $x_1 x_2 x_3 x_4 = c^4$.Since the equations are symmetric with respect to $x$ and $y$,by symmetry,$y_1 + y_2 + y_3 + y_4 = 0$ and $y_1 y_2 y_3 y_4 = c^4$.Thus,all the given options are correct.

If the circle $x^2 + y^2 = a^2$ intersects the hyperbola $xy = c^2$ in four points $P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3), S(x_4, y_4)$,then:

Similar Questions

Find the equations of the transverse axis and conjugate axis of the hyperbola $16x^2 - y^2 + 64x + 4y + 44 = 0$.

If the equation of the tangent to the hyperbola $5x^2 - 9y^2 - 20x - 18y - 34 = 0$ which makes an angle of $45^{\circ}$ with the positive $X$-axis is $x + by + c = 0$,then $b^2 + c^2 =$

If $L_1=0$ and $L_2=0$ are the asymptotes of the hyperbola $9x^2-4y^2+36x+8y-4=0$,then the product of the perpendicular distances from the point $(1,1)$ to the lines $L_1=0$ and $L_2=0$ is

The vertices of a hyperbola are at $(0, 0)$ and $(10, 0)$ and one of its foci is at $(18, 0)$. The equation of the hyperbola is:

If a hyperbola has asymptotes $3x - 4y - 1 = 0$ and $4x - 3y - 6 = 0$,then the transverse and conjugate axes of that hyperbola are

Vedclass Test Series

Exam Paper Generator

Online Exam Module