Exercise 20.6
Page no 20.50
Type 1
Question 1
Find the equation of the set of all points $P(x, y)$ such that the line $O P$ is coincident with the line joining $P$ and the point $(3,2)$.
Question 2
Find the equation of the set of points equidistant from $(-1,-1)$ and $(4,2)$.
Question 3
Find the equation of the locus of a point P . If the sum of squares of distance of the point P from the axes is $p^{2}$
Question 4
Find the equation of the set of all points which are equidistant from the points $\left(a^{2}+b^{2}, a^{2}-b^{2}\right)$ and $\left(a^{2}-b^{2}, a^{2}+b^{2}\right)$
Question 5
Square of the distance of the point from $x$-axis is double of its distance from the origin.
distance from $y$-axis is always
Question 6
Write the equation of locus of a point whose distance from $y$-axis is always equal to the double of its distance from $x$-axis.
Question 7
Find the equation of locus of a point whose distance from y-axis is always equal to the double of its distance from x-axis
Question 8
If a point $P$ moves such that its distance from $(a, 0)$ is always equal to $a+x$-coordinates of $P$, show that the locus of $P$ is $y^{2}=4 a x$.
Question 9
Show that the equation of the locus of a point which moves so that the sum of its distance from two given points $(k, 0)$ and $(-k, 0)$ is equal to $2 a$ is
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}-k^{2}}=1$
Question 10
If the sum of the distances of a moving point from two fixed points $(a e, 0)$ and $(-a e, 0)$ be $2 a$, prove that the locus of the point is
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}\left(1-e^{2}\right)}=1$
Question 11
Find the locus of a variable point $\left(a t^{2}, 2 a t\right)$, where $t$ is the parameter.
Question 12
If the coordinates of a variable point $P$ be $\left(t+\frac{1}{t}, t-\frac{1}{t}\right)$, where $t$ is a variable quantity, then find the locus of $P$.
Question 13
If the coordinates of a variable point $P$ be $(\cos \theta+\sin \theta, \sin \theta-\cos \theta)$, where $\theta$ is a variable quantity, find the locus of $P$.
Question 14
If $A(\cos \theta, \sin \theta), B(\sin \theta, \cos \theta), C(1,2)$ are the vertices of a $\triangle A B C$. Find the locus of its centroid if $\theta$ varies.
Question 15
A point moves so that its distance from the point $(-2,3)$ is always three times its distance from the point $(0,3)$. Find the equation to its locus.
Question 16
$A$ and $B$ are two given points whose coordinates are $(-5,3)$ and (2,4) respectively. A point $P$ moves in such a manner that $P A: P B=3: 2$. Find the equation to the locus traced out by $P$.
Question 17
Find the equation of the locus of points such that the sum of its distances from $(0,3)$ and $(0,-3)$ is $8 .$
Page no 20.50
Question 18
$S$ is the point $(4,0)$ and $M$ is the foot of the perpendicular drawn from a point $p$ to the $y$-axis. If $P$ moves such that the distance $P S$ and $P M$ remain equal, find the locus of $P$.
Question 19
If $A(\mathrm{l}, 1)$ and $B(-2,3)$ are two fixed points, find the locus of a point $P$ so that area of $\triangle P A B$ is 9 units.
Question 20
Find the locus of a point such that the line segment having end points $(2,0)$ and $(-2,0)$ subtend a right angle at that point.
Question 21
If $P$ is the middle point of the straight line joining a given point $A(1,2)$ and $Q$ where $Q$ is a variable point on the curve $x^{2}+y^{2}+x+y=0$. Find the locus of $P$.
Question 22
$A(2,3)$ is a fixed point and $Q(3 \cos \theta, 2 \sin \theta)$ a variable point. If $P$ divides $A Q$ internally in the ratio $3: 1$, find the locus of $P$.
Question 23
From the point $A(6,-8)$ all possible lines are drawn to cut the $x$-axis. Find the locus of their middle points.
Question 24
A stick of length $l$ slides with its ends on two mutually perpendicular lines. Find the locus of the middle point of the stick.
Question 25
Prove that the locus of the point equidistant from two given points is the straight line which bisects the line segment joining the given points at right angles.
Type 2
Question 26
Describe the locus of the point $(x, y)$ satisfying the condition $x^{2}+y^{2}=a^{2}$.
Question 27
Describe the locus of the point $(x, y)$ satisfying $(x-1)^{2}+(y-1)^{2}=2^{2}$.
Type 2
Question 28
Examine whether point $(1,2)$ lies on the curve $4 x^{2}-y^{2}=0$.
Question 29
Examine whether point $(2,-3)$ lies on the curve
$x^{2}-2 y^{2}+6 x y+8=0$
Question 30
If the equations $a x^{2}+2 h x y+b y^{2}=0$ and $b x^{2}-2 h x y+a y^{2}=0$ represent the same curve, then show that $a+b=0$.
Question 31
Find the value of $k$ if the point $(1,2)$ lies on the curve
$(k-10) x^{2}+y^{2}-(k-7) x-(3 k-27) y+11=0$
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