The equation of the locus of the centroid of | vedclass

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

Question

(A) Let the vertices of the triangle be $A = (a \cos k, a \sin k)$,$B = (b \sin k, -b \cos k)$,and $C = (1, 0)$.Let $G(x, y)$ be the centroid of the t

Vedclass · Accepted Answer

$(1-3x)^2 + 9y^2 = a^2 + b^2$

Vedclass · Answer

(A) Let the vertices of the triangle be $A = (a \cos k, a \sin k)$,$B = (b \sin k, -b \cos k)$,and $C = (1, 0)$.Let $G(x, y)$ be the centroid of the triangle.The coordinates of the centroid are given by $x = \frac{a \cos k + b \sin k + 1}{3}$ and $y = \frac{a \sin k - b \cos k + 0}{3}$.From these,we have $3x - 1 = a \cos k + b \sin k$ ... $(i)$ and $3y = a \sin k - b \cos k$ ... $(ii)$.Squaring and adding equations $(i)$ and $(ii)$:$(3x - 1)^2 + (3y)^2 = (a \cos k + b \sin k)^2 + (a \sin k - b \cos k)^2$.$(3x - 1)^2 + 9y^2 = a^2(\cos^2 k + \sin^2 k) + b^2(\sin^2 k + \cos^2 k) + 2ab \cos k \sin k - 2ab \sin k \cos k$.Since $\sin^2 k + \cos^2 k = 1$,we get $(3x - 1)^2 + 9y^2 = a^2 + b^2$.This is equivalent to $(1 - 3x)^2 + 9y^2 = a^2 + b^2$.

The equation of the locus of the centroid of the triangle whose vertices are $(a \cos k, a \sin k)$,$(b \sin k, -b \cos k)$ and $(1, 0)$,where $k$ is a parameter,is

Similar Questions

Let $A = (a, 0)$ and $B = (-a, 0)$ be two fixed points. For $a \in (-\infty, 0)$,point $P$ moves in the plane such that $PA = nPB$ $(n \neq 0, 1)$. If the locus of $P$ is a circle,then the circle:

$A$ circle touches the $X$-axis and also touches the circle with radius $2$ and center at $(0, 3)$. Find the locus of the center of the circle.

The locus of the centre of the circles which touch both the circles $x^{2}+y^{2}=a^{2}$ and $x^{2}+y^{2}=4ax$ externally is

The locus of the midpoints of the chords of the circle $x^2+y^2=16$,which are tangents to the hyperbola $9x^2-16y^2=144$,is

If a circle of a constant radius $6$ passes through the origin $O$ and meets the coordinate axes at $A$ and $B$, then find the locus of the centroid of $\triangle OAB$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module