The locus of a point which moves such that th | vedclass

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

Question

(B) Let the vertices of the triangle be $A(x_1, y_1)$,$B(x_2, y_2)$,and $C(x_3, y_3)$.Let the moving point be $P(x, y)$.The sum of the squares of the

Vedclass · Accepted Answer

Centroid of the triangle

Vedclass · Answer

(B) Let the vertices of the triangle be $A(x_1, y_1)$,$B(x_2, y_2)$,and $C(x_3, y_3)$.Let the moving point be $P(x, y)$.The sum of the squares of the distances from $P$ to the vertices is constant,say $K$:$(x - x_1)^2 + (y - y_1)^2 + (x - x_2)^2 + (y - y_2)^2 + (x - x_3)^2 + (y - y_3)^2 = K$Expanding the terms:$3x^2 - 2x(x_1 + x_2 + x_3) + 3y^2 - 2y(y_1 + y_2 + y_3) + (x_1^2 + x_2^2 + x_3^2 + y_1^2 + y_2^2 + y_3^2) = K$Dividing by $3$:$x^2 - 2x\left(\frac{x_1 + x_2 + x_3}{3}ight) + y^2 - 2y\left(\frac{y_1 + y_2 + y_3}{3}ight) = 	ext{constant}$This is the equation of a circle with centre at $\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}ight)$,which is the centroid of the triangle.

The locus of a point which moves such that the sum of the squares of its distances from the three vertices of a triangle is constant,is a circle whose centre is at the

Similar Questions

The locus of the centres of the circles,which touch the circle $x^2 + y^2 = 1$ externally,also touch the $y$-axis and lie in the first quadrant is

If a circle passes through the point $(a, b)$ and cuts the circle ${x^2} + {y^2} = 4$ orthogonally,then the locus of its centre is

If the line $lx + my = 1$ is a tangent to the circle ${x^2} + {y^2} = {a^2}$,then the locus of the point $(l, m)$ is

For any value of $\theta$,if the straight lines $x \sin \theta + (1 - \cos \theta) y = a \sin \theta$ and $x \sin \theta - (1 + \cos \theta) y + a \sin \theta = 0$ intersect at $P(\theta)$,then the locus of $P(\theta)$ is a

If one end of the diameter is $(1, 1)$ and the other end lies on the line $x+y=3$,then the locus of the centre of the circle is

Vedclass Test Series

Exam Paper Generator

Online Exam Module