Locus of a Point

The sum of the distance of a moving point from the points $(4,0)$ and $(-4,0)$ is always $10$ units. Find the equation of the locus of the moving point.

Find the points on the locus of points that are $3$ units from $x-$axis and $5$ units from the point $(5,1).$

If $R$ is any point on the$x-$axis and $Q$ is any point on the $y-$axis and $P$ is a variable point on $RQ$ with $RP=b , PQ=a.$ then find the equation of locus of $P$.

Find the value of $K$ and $b,$ if the points $P(-3,1)$ and $Q(2,b)$ lie on the locus of $x^2-5x+ky=0.$

Straight line joining the points $(2,3)$ and $(-1,4)$ passes through the point $(\alpha , \beta )$ if

If the point $(8, -5)$ lies on the locus $\frac{x^2}{16}-\frac{y^2}{25}=k$, then the value of $k$ is

Which of the following point lie on the locus of $3x^2+3y^2-8x-12y+17=0$

Which of the following equation is the locus of $(a{t}^2,2at)$

The equation of the locus of the point whose distance from $y-$axis is half the distance from origin is

If $Q$ is a point on the locus of $x^2+y^2+4x-3y+7=0$, If the equation of locus of $P$ which divides segment $OQ$ externally in the ratio $3:4,$is $x^2+y^2-12x+9y+k=0$, then find the value of $k$.

If the points $P(6,2)$ and $Q(-2,1)$ and $R$ are the vertices of a $\Delta PQR$ and R is the point on the locus $y=x^2-3x+4,$ then find the value of $\lambda$ if the equation of the locus of centroid of $\Delta PQR$ is $9x^2-33x-3y+\lambda =0$.

If $P(2,-7)$ is a given point and $Q$ is a point on $2x^2+9y^2=18,$ then find the value of $k$, if the equations of the locus of the mid-point of $PQ$ is $8x^2+36y^2-16x+252y+k=0$.

The coordinates of a moving point $P$ are $\left(\frac{a}{2}(\cos ec\theta +\sin \theta ) , \frac{b}{2}(\cos ec\theta -\sin \theta )\right),$ where $\theta$ is a variable parameter. Show that the equation of the locus $P$ is $b^2{x}^2-a^2{y}^2=a^2{b}^2$.

If $O$ is origin and $R$ is a variable point on $y^2=4x,$ then find the value of $k$, if the equation of the locus of the midpoint of the line segment $OR$ is $y^2=kx$.

Find the value of $k$. If the equation of the locus of the point $P$ such that the line segment AB, joining the points$A(1,-6)$ and $B(4,-2),$ subtends a right angle at $P$ is given by $x^2+y^2-5x+8y+k=0$.

Find the value of $k$, if the equation of the locus of a point such that the sum of the squares of the distance from the points $(3,5),(1,-1)$ equal to $20$ is $x^2+y^2-4x-4y+k=0$

A straight rod of length $8$ units slides with its ends $A$ and $B$ always on the $x$ and $y$ axes respectively. Find the value of $\lambda$, if the locus of the midpoint of the line segment $AB$ is $x^2+y^2=\lambda$.

If $\theta$ is a parameter and the equation of the locus of a moving point, whose coordinates are $x=a{\cos }^3\theta , y=a{\sin }^3\theta$ is $x^k+y^k=a^k$. Find the value of $k$.

Find the value of $k$, if the locus of a point $P$ that moves at a constant distance of three units and parallel to right side of $y$-axis is $x=\pm k$

Find the locus of a point $P$ that moves at a constant distance of two units above the $x-$axis.