The centres of those circles which touch the | vedclass

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

Question

(B) The given circle is $x^{2} + y^{2} - 8x - 8y - 4 = 0$.The center of this circle is $(4, 4)$ and its radius $r = \sqrt{4^{2} + 4^{2} - (-4)} = \sqr

Vedclass · Accepted Answer

a parabola

Vedclass · Answer

(B) The given circle is $x^{2} + y^{2} - 8x - 8y - 4 = 0$.The center of this circle is $(4, 4)$ and its radius $r = \sqrt{4^{2} + 4^{2} - (-4)} = \sqrt{16 + 16 + 4} = \sqrt{36} = 6$.Let the center of the required circle be $(h, k)$ and its radius be $R$. Since the circle touches the $x$-axis,its radius $R = |k|$. Assuming the circle is above the $x$-axis,$R = k$.Since the circle touches the given circle externally,the distance between their centers is equal to the sum of their radii:$\sqrt{(h - 4)^{2} + (k - 4)^{2}} = 6 + k$.Squaring both sides:$(h - 4)^{2} + (k - 4)^{2} = (6 + k)^{2}$.Expanding the terms:$h^{2} - 8h + 16 + k^{2} - 8k + 16 = 36 + k^{2} + 12k$.Simplifying the equation:$h^{2} - 8h - 20k - 4 = 0$.Replacing $(h, k)$ with $(x, y)$,we get the locus:$x^{2} - 8x - 20y - 4 = 0$.This is the equation of a parabola.

The centres of those circles which touch the circle $x^{2} + y^{2} - 8x - 8y - 4 = 0$ externally and also touch the $x$-axis,lie on:

Similar Questions

The locus of the centers of the circles touching the lines $3x - 4y + 1 = 0$ and $12x + 5y - 1 = 0$ is/are:

$A$ point $P$ moves in such a way that the ratio of its distance from two coplanar points is always a fixed number $(\lambda \ne 1)$. Then its locus is

The locus of the midpoints of the chords of the circle $x^{2}+y^{2}=4$ which subtend a right angle at the origin is

What does the equation $\sqrt{(x - 2)^2 + (y + 1)^2} + \sqrt{(x + 2)^2 + (y + 4)^2} = 5$ represent?

The equation to the locus of a point which moves so that its distance from the $x$-axis is always one-half its distance from the origin is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module