Exercise 24.1
Page no 24.29
Type 1
Question 1
Find the equation to the ellipse whose
(i) one focus is $(-1,1)$. directrix is $x-y+3=0$ and eccentricity is $\frac{1}{2}$ -
(ii) one focus is $(6,7)$, directrix is $x+y+2=0$ and eccentricity is $\frac{1}{\sqrt{3}}$.
Question 2
Find the equation to the ellipse whose one focus is $(2,1)$, the directrix is $2 x-y+3=0$ and the eccentricity is $\frac{1}{\sqrt{2}}$.
Question 3
Find the equation of the ellipse with center at the origin, the length of the major axis 12 and one focus at $(4,0)$.
Question 4
Find the equation to the ellipse whose center is $(-2,3)$ and whose semi-axes are 3 and 2 when the major axis is parallel to the $y$ axis.
Question 5
Find the equation of the ellipse having vertices $(0, \pm 10)$ and eccentricity $\frac{4}{5}$.
Question 6
Find the equation of the ellipse
(i) having vertices at $(0, \pm 13)$ and foci at $(0, \pm 5)$
(ii) having vertices at $(\pm 5,0)$ and foci $(\pm 4,0)$
(iii) having vertices at $(\pm 13,0)$ and foci $(\pm 5,0)$
(iv) having vertices at $(\pm 6,0)$ and foci $(\pm 4,0)$.
Question 7
Find the equation of the ellipse having
(i) length of major axis 26 and foci $(\pm 5,0)$
(ii) length of major axis 20 and foci $(0, \pm 5)$
(iii) length of major axis 8 and foci $(\pm 3,0)$
(iv) length of minor axis 16 and foci $(0, \pm 6)$
Question 8
Find the equation of the ellipse passing through the point $(3,2)$, having center $(0,0)$ and major axis on y-axis.
Question 9
Find the equation of the ellipse having ends of major axis $(0, \pm \sqrt{5})$ and ends of minor axis $(\pm 1,0)$.
Question 10
If $a$ be the length of semi major axis, $b$ the length semi minor axis and $c$ the distance of one focus from the center of an ellipse, then find the equation of the ellipse for which center is $(0,0)$ foci on $x$-axis, $b=3$ and $c=4$.
Question 11
The distance between the foci of an ellipse is 10 and its latus rectum is 15: find its equation referred to its axes as axes of coordinates.
Question 12
Find the equation of the ellipse in the standard form whose minor axis is equal to the distance between the foci and whose latus rectum is 10 .
Question 13
The eccentricity of an ellipse is $\frac{1}{2}$ and the distance between its foci is 4 units. If the major and minor axes of the ellipse are respectively along the $x$ and $y$ axes, find the equation of the ellipse.
Page no 24.30
Question 14
Find the equation of the ellipse passing through $(6,4)$, foci on y-axis center at the origin and having eccentricity $\frac{3}{4}$.
Question 15
Find the equation of the ellipse passing through $(4,1)$ with focus as $(\pm 3,0)$.
Question 16
Find the equation of the set of all points whose distances from $(0,4)$ are $\frac{2}{3}$ of their distances from the line $y=9$.
Question 17
Find the equation to the ellipse whose foci are $(4,0)$ and $(-4,0)$ and eccentricity is $\frac{1}{3}$.
Question 18
Find the equation of the ellipse, referred to its axes as the $x, y$ axes respectively, which passes through the point $(-3,1)$ and has the eccentricity $\sqrt{\frac{2}{3}}$.
Question 19
If the angle between the lines joining the foci of any ellipse to an extremity of the minor axis is $90^{\circ}$, find the eccentricity. Find also the equation of the ellipse if the major axis is $2 \sqrt{2}$.
Type 2
Question 20
For the ellipse $9 x^{2}+16 y^{2}=144$, find the lengths of the major and minor axes, the eccentricity, the coordinates of the foci, the vertices and the equations of the directrices.
Question 21
Find the lengths of the major and the minor axes, the coordinates of the foci, the vertices, the eccentricity, the length of latus rectum and the equation of the directrices of the following ellipses :
(i) $\frac{x^{2}}{169}+\frac{y^{2}}{25}=1$
(ii) $\frac{x^{2}}{25}+\frac{y^{2}}{169}=1$
(iii) $3 x^{2}+2 y^{2}=18$
(iv) $x^{2}+16 y^{2}=16$
Question 22
Find the coordinates of the foci, the vertices the length of major axis , the minor axis , the eccentricity and the length of latus rectum of the following ellipses:
(i) $4 x^{2}+9 y^{2}=36$
(ii) $9 x^{2}+4 y^{2}=36$
(iii) $36 x^{2}+4 y^{2}=144$
(iv) $25 x^{2}+4 y^{2}=100$
(v) $\frac{x^{2}}{100}+\frac{y^{2}}{400}=1$
(vi) $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$
(vii) $\frac{x^{2}}{25}+\frac{y^{2}}{100}=1$
(viii) $\frac{x^{2}}{49}+\frac{y^{2}}{36}=1$
(ix) $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$
Question 23
Show that the following equation represents an ellipses and find its center and eccentricity:
$8 x^{2}+6 y^{2}-16 x+12 y+13=0
Question 24
Find the center, the lengths of the axes and the eccentricity of the ellipse
$2 x^{2}+3 y^{2}-4 x-12 y+13=0$
Page no 24.31
Question 25
Show that $4 x^{2}+y^{2}-8 x+2 y+1=0$ Represent an ellipse. Find its eccentricity.
Question 26
find the latus rectum, eccentricity, coordinates of the foci and the length of axis of the following ellipses
(a) $2 x^{2}+5 y^{2}-30 y=0$
(ii) $4 x^{2}+9 y^{2}-8 x-36 y+4=0$
Type 3
Question 27
Find the eccentricity of an ellipse if tis latus rectum is equal to one-half of its major axis
Question 28
Find the eccentricity of an ellipse if its latus rectum is one third of its major axis
Question 29
A rod AB of length 15 cm rests in between two coordinate axes in such a way that the point A lies on x-axis and end point B lies on x-axis A point p (x,y) is taken on the rod in such a way that AP = 6cm . Show that the locus of P is an ellipse $\frac{x^{2}}{81}+\frac{y^{2}}{36}=1$
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