The locus of the centre of a variable circle | vedclass

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

Question

(D) Let the equation of the variable circle be $x^2 + y^2 + 2gx + 2fy + c = 0$.Two circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2

Vedclass · Accepted Answer

$x - y = 0$

Vedclass · Answer

(D) Let the equation of the variable circle be $x^2 + y^2 + 2gx + 2fy + c = 0$.Two circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ cut orthogonally if $2g_1g_2 + 2f_1f_2 = c_1 + c_2$.For the first circle $x^2 + y^2 - 2x - 4y - 1 = 0$,we have $g_1 = -1, f_1 = -2, c_1 = -1$. The condition is $2g(-1) + 2f(-2) = c - 1$,which simplifies to $-2g - 4f = c - 1$  .......$(1)$For the second circle $x^2 + y^2 - 4x - 2y - 1 = 0$,we have $g_2 = -2, f_2 = -1, c_2 = -1$. The condition is $2g(-2) + 2f(-1) = c - 1$,which simplifies to $-4g - 2f = c - 1$  .......$(2)$Subtracting equation $(2)$ from equation $(1)$:$(-2g - 4f) - (-4g - 2f) = (c - 1) - (c - 1)$$2g - 2f = 0$$g - f = 0$Since the centre of the variable circle is $(-g, -f)$,let the centre be $(x, y)$. Thus $x = -g$ and $y = -f$.Substituting $g = -x$ and $f = -y$ into $g - f = 0$,we get $-x - (-y) = 0$,which simplifies to $x - y = 0$.

The locus of the centre of a variable circle which cuts the circles $x^2 + y^2 - 2x - 4y - 1 = 0$ and $x^2 + y^2 - 4x - 2y - 1 = 0$ orthogonally is:

Similar Questions

The locus of the vertex $A$ of a triangle $ABC$,where the base $BC$ is fixed and the perimeter of the triangle is constant,is

If a tangent to the circle $x^2 + y^2 = 1$ intersects the coordinate axes at distinct points $P$ and $Q,$ then the locus of the mid-point of $PQ$ is

The two points $A$ and $B$ in a plane are such that for all points $P$ lying on a circle satisfying $\frac{PA}{PB} = k$,then $k$ will not be equal to:

The locus of the midpoints of the chords of the circle $x^2-2x+y^2=0$ drawn from the point $(0,0)$ on it is

The set of all points that are at a distance of at least $2$ units from $(-3, 0)$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module