A rhombus is inscribed in the region common t | vedclass

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

Question

(A) The given circles are $C_1: x^2 + y^2 - 4x - 12 = 0$ (centre $(2, 0)$,radius $r = \sqrt{2^2 + 0^2 + 12} = 4$) and $C_2: x^2 + y^2 + 4x - 12 = 0$ (

Vedclass · Accepted Answer

$8\sqrt{3}$ sq.units

Vedclass · Answer

(A) The given circles are $C_1: x^2 + y^2 - 4x - 12 = 0$ (centre $(2, 0)$,radius $r = \sqrt{2^2 + 0^2 + 12} = 4$) and $C_2: x^2 + y^2 + 4x - 12 = 0$ (centre $(-2, 0)$,radius $r = 4$).The line joining the centres is the $x$-axis $(y = 0)$.The common chord is obtained by subtracting the equations: $(x^2 + y^2 - 4x - 12) - (x^2 + y^2 + 4x - 12) = 0$ $\Rightarrow -8x = 0$ $\Rightarrow x = 0$ (the $y$-axis).The vertices of the rhombus on the $x$-axis are the centres of the circles,$(-2, 0)$ and $(2, 0)$,so the length of one diagonal is $d_1 = 2 - (-2) = 4$.The other diagonal lies on the common chord $x = 0$. The intersection points of $x = 0$ with $x^2 + y^2 - 4x - 12 = 0$ are $y^2 - 12 = 0 \Rightarrow y = \pm 2\sqrt{3}$.Thus,the length of the second diagonal is $d_2 = 2\sqrt{3} - (-2\sqrt{3}) = 4\sqrt{3}$.The area of the rhombus is $A = \frac{1}{2} 	imes d_1 	imes d_2 = \frac{1}{2} 	imes 4 	imes 4\sqrt{3} = 8\sqrt{3}$ sq.units.

$A$ rhombus is inscribed in the region common to the two circles $x^2 + y^2 - 4x - 12 = 0$ and $x^2 + y^2 + 4x - 12 = 0$,with two of its vertices on the line joining the centres of the circles. The area of the rhombus is:

Similar Questions

The sum of diameters of the circles that touch $(i)$ the parabola $75x^2 = 64(5y - 3)$ at the point $\left(\frac{8}{5}, \frac{6}{5}\right)$ and $(ii)$ the $y$-axis,is equal to $......$

If $P(2, 8)$ is an interior point of a circle $x^2 + y^2 - 2x + 4y - p = 0$ which neither touches nor intersects the axes,then the set for $p$ is

The equations of the circles which touch the lines $3x - 4y + 1 = 0$ and $4x + 3y - 7 = 0$ and pass through the point $(2, 3)$ are:

Two circles each of radius $5$ units touch each other at $(1,2)$ and $4x+3y=10$ is their common tangent. The equation of that circle among the two given circles,such that some portion of it lies in every quadrant is

The values of the constant term in the equation of a circle passing through $(1, 2)$ and $(3, 4)$ and touching the line $3x + y - 3 = 0$ are

Vedclass Test Series

Exam Paper Generator

Online Exam Module