If the circle x^2 + y^2 + 2x - 4y - k = 0 lie | vedclass

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

Question

(C) The given equations of the circles are of the form $x^2 + y^2 + 2x - 4y + C = 0$.These circles are concentric with the center $(-g, -f) = (-1, 2)$

Vedclass · Accepted Answer

$11$

Vedclass · Answer

(C) The given equations of the circles are of the form $x^2 + y^2 + 2x - 4y + C = 0$.These circles are concentric with the center $(-g, -f) = (-1, 2)$.The radius $r$ of a circle $x^2 + y^2 + 2x - 4y + C = 0$ is given by $r = \sqrt{g^2 + f^2 - C} = \sqrt{1 + 4 - C} = \sqrt{5 - C}$.Let the radii of the three circles be $r_1, r_2,$ and $r_3$ respectively.For the first circle,$C_1 = -4$,so $r_1^2 = 5 - (-4) = 9$,which means $r_1 = 3$.For the third circle,$C_3 = -20$,so $r_3^2 = 5 - (-20) = 25$,which means $r_3 = 5$.Since the middle circle lies exactly between the other two,its radius $r_2$ must be the arithmetic mean of $r_1$ and $r_3$.$r_2 = \frac{r_1 + r_3}{2} = \frac{3 + 5}{2} = 4$.Now,for the middle circle,$r_2^2 = 5 - (-k) = 5 + k$.Since $r_2 = 4$,$r_2^2 = 16$.Therefore,$5 + k = 16$,which gives $k = 11$.

If the circle $x^2 + y^2 + 2x - 4y - k = 0$ lies exactly between the circles $x^2 + y^2 + 2x - 4y - 4 = 0$ and $x^2 + y^2 + 2x - 4y - 20 = 0$,then $k = \dots$

Similar Questions

The gradient of the radical axis of the circles ${x^2} + {y^2} - 3x - 4y + 5 = 0$ and $3{x^2} + 3{y^2} - 7x + 8y + 11 = 0$ is

The distance between the centres of similitude of the circles $x^2+y^2+6x-8y+16=0$ and $x^2+y^2-2x-2y+1=0$ is

Suppose the circle $S: x^2+y^2+2gx+2fy+c=0$ cuts orthogonally the two circles $S': x^2+y^2-4x-6y+11=0$ and $S'': x^2+y^2-10x-4y+21=0$. If the centre of $S=0$ lies on the bisector of the angle between the positive coordinate axes,then $2g+2f+c=$

The equation of the circle passing through the points of intersection of two circles $x^2+y^2+2x+3y+1=0$ and $x^2+y^2+4x+3y+2=0$ and the point $(-1,1)$ is

$A$ circle $C$ passes through $(2a, 0)$ and the line $2x = a$ is the radical axis of the circle $C$ and the circle $x^2 + y^2 = a^2$. Then,

Vedclass Test Series

Exam Paper Generator

Online Exam Module