The coordinates of the points O,A,and B are ( | vedclass

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

Question

(A) Let the coordinates of the moving point $P$ be $(x, y)$.The coordinates of the given points are $O(0, 0)$,$A(0, 4)$,and $B(6, 0)$.The area of a tr

Vedclass · Accepted Answer

$(x - 3y)(x + 3y) = 0$

Vedclass · Answer

(A) Let the coordinates of the moving point $P$ be $(x, y)$.The coordinates of the given points are $O(0, 0)$,$A(0, 4)$,and $B(6, 0)$.The area of a triangle with vertices $(x_1, y_1)$,$(x_2, y_2)$,and $(x_3, y_3)$ is given by $\frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$.Area of $\Delta POA = \frac{1}{2} |0(4 - y) + 0(y - 0) + x(0 - 4)| = \frac{1}{2} |-4x| = 2|x|$.Area of $\Delta POB = \frac{1}{2} |0(0 - y) + 6(y - 0) + x(0 - 0)| = \frac{1}{2} |6y| = 3|y|$.According to the problem,Area of $\Delta POA = 2 	imes$ Area of $\Delta POB$.Therefore,$2|x| = 2 	imes 3|y|$.$|x| = 3|y|$.Squaring both sides,we get $x^2 = 9y^2$,which implies $x^2 - 9y^2 = 0$.This can be factored as $(x - 3y)(x + 3y) = 0$.

The coordinates of the points $O$,$A$,and $B$ are $(0,0)$,$(0,4)$,and $(6,0)$ respectively. If a point $P$ moves such that the area of $\Delta POA$ is always twice the area of $\Delta POB$,then the equation to both parts of the locus of $P$ is

Similar Questions

$O$ is the origin and $A$ is the point $(3,4)$. If a point $P(x,y)$ moves such that the line segment $OP$ is always parallel to the line segment $OA$,then the equation of the locus of $P$ is:

$A$ line is at a constant distance $c$ from the origin and meets the coordinate axes in $A$ and $B$. The locus of the centre of the circle passing through $O, A, B$ is

$A$ moving line intersects the lines $x+y=0$ and $x-y=0$ at the points $A$ and $B$ respectively, such that the area of the triangle with vertices $(0,0)$, $A$, and $B$ has a constant area $C$. The locus of the mid-point of $AB$ is given by the equation:

Let $P(2, -3)$ and $Q(-2, 1)$ be the vertices of the $\Delta PQR$. If the centroid of $\Delta PQR$ lies on the line $2x + 3y = 1$, then the locus of $R$ is

If the sum of the distances of a point $P(x, y)$ from two perpendicular lines in a plane is $1$,then the locus of $P$ is a

Vedclass Test Series

Exam Paper Generator

Online Exam Module