The locus of the centre of a circle which cut | vedclass

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

Question

(D) Let the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$ $(i)$.It cuts the circle $x^2 + y^2 - 20x + 4 = 0$ orthogonally.The condition for orthogonality

Vedclass · Accepted Answer

$y^2 = 16x$

Vedclass · Answer

(D) Let the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$ $(i)$.It cuts the circle $x^2 + y^2 - 20x + 4 = 0$ orthogonally.The condition for orthogonality is $2g_1g_2 + 2f_1f_2 = c_1 + c_2$.Here,$2g(-10) + 2f(0) = c + 4$,which gives $-20g = c + 4$ $(ii)$.Circle $(i)$ touches the line $x = 2$,so the perpendicular distance from the centre $(-g, -f)$ to the line $x - 2 = 0$ equals the radius $\sqrt{g^2 + f^2 - c}$.$|(-g) - 2| = \sqrt{g^2 + f^2 - c} \Rightarrow (g + 2)^2 = g^2 + f^2 - c$.$g^2 + 4g + 4 = g^2 + f^2 - c \Rightarrow 4g + 4 = f^2 - c$ $(iii)$.Adding $(ii)$ and $(iii)$,we get $-20g + 4g + 4 = c + 4 + f^2 - c$.$-16g + 4 = f^2 + 4 \Rightarrow f^2 = -16g$.Replacing $(-g, -f)$ with $(x, y)$,we have $g = -x$ and $f = -y$.Substituting these,$(-y)^2 = -16(-x) \Rightarrow y^2 = 16x$.

The locus of the centre of a circle which cuts orthogonally the circle $x^2 + y^2 - 20x + 4 = 0$ and which touches the line $x = 2$ is:

Similar Questions

If $t$ is a parameter,$A = (a \sec t, b \tan t)$,$B = (-a \tan t, b \sec t)$,and $O = (0, 0)$,then the locus of the centroid of $\triangle OAB$ is:

$A$ point $P(x, y)$ divides the line segment joining the points $(5, 0)$ and $(10 \cos \theta, 10 \sin \theta)$ in the ratio $2 : 3$. Find the locus of point $P$ as $\theta$ varies.

If a point $P$ moves such that the sum of the distances from $P$ to the points $A(1, -1)$ and $B(-1, 1)$ is always $4$,then the equation for the locus of $P$ is

$A$ point moves in such a way that its distance from the origin is always $4$. Then the locus of the point is:

The locus of the centres of the circles,which cut the circles $x^2+y^2+4x-6y+9=0$ and $x^2+y^2-5x+4y+2=0$ orthogonally,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module