Exercise 23.1
Page no 23.27
Type 1
Question 1
(i) Find the equation of the parabola whose focus is $(0,-2)$ and directrix is $y=2$.
(ii) Find the equation of the parabola whose focus is $(0,-3)$ and directrix is $y=3$.
Question 2
(i) Find the equation of the parabola whose focus is $(4,0)$ and directrix is $x=-4$
(ii) Find the equation of the parabola whose focus is $(6,0)$ and directrix is $x=-6$
(iii) Find the equation of the parabola whose focus is $(2,0)$ and directrix is $x=-2$.
Question 3
Find the equation of the parabola whose focus is $(-1,2)$ and directrix is $x-2 y-15=0$
Question 4
Find the equation of the parabola whose focus is $(2,3)$ and directrix is $x-2 y-6=0$.
Question 5
Find the equation of the parabola passing through $(2,3)$ with vertex at the origin and axis along $x$-axis.
Question 6
Find the equation of the parabola whose focus is at $(-1,1)$ and the directrix is $x+y+1=0$
Question 7
Find the equation of the parabola whose focus is $(5,3)$ and directrix the line $3 x-4 y+1=0$
Question 8
Find the equation of the parabola if the focus is the point $\left(\frac{5}{4},-1\right)$ and the directrix is $4 x-13=0$
Question 9
(i) Find the equation of the parabola having the vertex at $(0,1)$ and the focus at $(0,0)$.
(ii) Find the equation of the parabola with vertex at $(0,0)$ and focus at $(0,2)$.
Question 10
(i) Find the equation of the parabola if the focus is at $(-6,-6)$ and vertex is at $(-2,2)$
(ii) Find the equation of the parabola whose focus is $(3,0)$ and vertex is $(0,0)$.
(iii) Find the equation of the parabola having vertex at $(0,0)$ and focus at $(-2,0)$.
Page no 23.28
Question 11
(i) Find the equation of the parabola passing through $(5,2)$. having vertex a 16 : and symmetric about y-axis.
(ii) Find the equation of the parabola passing through $(2,3)$ axis along x- axis and vertex at $(0,0)$.
(iii) Find the equation of the parabola whose focus is at $(0,0)$ and vertex is at the intersection of the line $x+y=1$ and $x-y=3$.
Question 12
Prove that the equation of the parabola whose focus is $(0,0)$ and tangent at the vertex is $x-y+1=0$ is $x^{2}+y^{2}+2 x y-4 x+4 y-4=0$
Question 13
Find the equation of the parabola whose vertex is at $(2,1)$ and the directrix is $x=y-1$
Question 14
Find the equation to the parabola whose axis is parallel to y- axis and which passes through the points $(0,4),(1,9)$ and $(-2,6)$ and determine its latus rectum
Question 15
Find the equation of the parabola which has its axis along the x-axis and what passes through the points $(3,2)$ and $(-2,-1)$.
Question 16
Find the equation of the parabola the extremities of whose latus rectum at: $(3,5)$ and $(3,-3)$.
Question 17
Find the equation of the parabola with its axis parallel to x-axis and which passes through the points $(1,2),(-1,3)$ and $(-2,1)$.
Type 2
Question 18
For each of the following parabolas, find the coordinates of the focus, the equation of the directrix and the length of latus rectum.
(i) $x^{2}=6 y$
(ii) $y^{2}=8 x$
(iii) $y^{2}=12 x$
(iv) $x^{2}=-9 y$
(v) $y^{2}=-12 x$
(vi) $y^{2}=10 x$
(vii) $y^{2}=-8 x$
(viii) $x^{2}=6 y$
Question 19
Find the vertex, focus, axis and latus rectum of the parabolas $4 y^{2}+12 x-20 y+67=0$
Question 20
0. Find the vertex, focus, axis, directrix and latus rectum of the following parabolas :
(i) $(y-2)^{2}=3(x+1)$.
(ii) $y^{2}+4 x+4 y-3=0$
Question 21
Find the vertex and the directrix of the parabola $y^{2}-3 x-2 y+7=0$.
Question 22
Find the focus and directrix of the parabola $3 x^{2}+12 x+8 y=0$
Question 23
Find the equation of the parabola with focus $(5,0)$ and directrix $x=-5$ Also, find the length of the latus rectum.
Question 24
Find the equation of the circle described on the line segment joining the fact of the parabolas $x^{2}=4 a y$ and $y^{2}=4 a(x-a)$ as diameter.
Type 3
Question 25
An equilateral triangle is inscribed in the parabola $y^{2}=4 a x$ whose vertex is at the vertex of the parabola. Find the length of its side.
Page no 23.29
Question 26
Find the area of the triangle formed by the lines joining the vertex of the parabola $x^{2}=12 y$ to the ends of its latus rectum.
Question 27
$P Q$ is a double ordinate of a parabola $y^{2}=4 a x$. Find the locus of its points of trisection.
Type 3
Question 28
The focus of a parabolic mirror as shown in the figure $23.46$ is at a distance of $6 \mathrm{~cm}$ from its vertex. If the mirror is $20 \mathrm{~cm}$ deep. Find the distance L.M.
(Image to be added)
Question 29
A water jet from a fountain reaches its maximum height of 4 meters at a distance of $0.5$ meter from the vertical passing through the point $O$ of the water outlet. Find the height of the jet above the horizontal $O X$ at a distance $0.75$ meter from the point $O$.
Question 30
The towers of a bridge, hung in the form of a parabola, have their tops 30 meters above the road-way and are 200 meters apart. If the cable is 5 meters above the roadway at the center of the bridge, find the length of the vertical supporting cable 30 meters from the center.
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