Examples

1. The equation $\frac{x^2}{10-a}+\frac{y^2}{4-a}=1$ represents an ellipse if

(a) $a<4$

(b) $a>4$

(c) $4<\mathrm{a}<10$

(d) $\mathrm{a}>10$

2. The radius of the circle passing through the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ and having its centre $(0,3)$ is

(a) 4

(b) 3

(c) $\sqrt{12}$

(d) $7/2$

3. The equation of the ellipse whose focus is $(1,-1)$ directrix $x-y-3=0$ and eccentricity $\frac{1}{2}$ is

(a) $7x^2+2xy+7y^2-10x+10y+7=0$

(c) $7x^2+2xy+7y^2+10x-10y-7=0$

(b) $7x^2+2xy+7y^2+7=0$

(d) None of these

4. The equation $(5x-1{)}^2+(5y-2{)}^2=({\lambda }^2-2\lambda +1)(3x+4y-1{)}^2$ represents an ellipse if $\lambda \in$.

(a) $(0,1)$

(b) $(0,2)-\{1\}$

(c) $(1,2)$

(d) $(-1,0)$

5. The eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ whose latus rectum is half of its major axis is

(a) $\frac{1}{\sqrt{2}}$

(b) $\sqrt{\frac{2}{3}}$

(c) $\frac{\sqrt{3}}{2}$

(d) None of these

6. If $(5,12)$ and $(24,7)$ are the foci of an ellipse passing through the origin, then the eccentricity of the conic is

(a) $\frac{\sqrt{386}}{12}$

(b) $\frac{\sqrt{386}}{13}$

(c) $\frac{\sqrt{386}}{25}$

(d) $\frac{\sqrt{386}}{38}$

Show Answer

Solution :

$\mathrm{S}(5,12),{\mathrm{S}}^{\prime }(24,7)$

$\begin{array}{rl}& \mathrm{SP}=\sqrt{5^2+{12}^2}=13\\ & \begin{array}{rl}& {\mathrm{S}}^{\prime }\mathrm{P}=\sqrt{{24}^2+7^2}=25\\ & {\mathrm{S}}^{\prime }=2\mathrm{ae}=\sqrt{{19}^2+5^2}\\ & =\sqrt{361+25}\\ & =\sqrt{386}\\ & =\frac{\sqrt{386}}{2}\\ & \mathrm{ae}\end{array}\\ & \begin{array}{rl}\mathrm{SP}+{\mathrm{S}}^{\prime }\mathrm{P}& =2\mathrm{a}=13+25\\ 2\mathrm{a}& =38\\ \mathrm{a}& =19\\ \mathrm{e}& =\frac{{\mathrm{SS}}^{\prime }}{\mathrm{SP}+{\mathrm{S}}^{\prime }\mathrm{P}}=\frac{2\mathrm{ae}}{2\mathrm{a}}\\ & =\frac{\sqrt{386}}{38}\end{array}\end{array}$

7. Locus of the point which divides double ordinate of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in the ratio 1:2 internally is

(a) $\frac{x^2}{a^2}+\frac{9y^2}{b^2}=1$

(b) $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{9{\mathrm{y}}^2}{{\mathrm{b}}^2}=\frac{1}{9}$

(c) $\frac{9x^2}{a^2}+\frac{9y^2}{b^2}=1$

(d) None of these.

Show Answer

Solution :

Let $\mathrm{P}(\mathrm{h},\mathrm{k})$ be a point divides double ordinate in the ratio $1:2$ internally

Let coordinates of ends of double ordinate $(h,y_1)$ and $(h,-y_1)$.

$\therefore$ By section formula $\mathrm{k}=\frac{-{\mathrm{y}}_1+2{\mathrm{y}}_1}{3}=\frac{{\mathrm{y}}_1}{3}$

${\mathrm{y}}_1=3\mathrm{k}$

Now the point $(\mathrm{h},{\mathrm{y}}_1)=(\mathrm{h},3\mathrm{k})$ lies on the ellipse

$\therefore \frac{{\mathrm{h}}^2}{{\mathrm{a}}^2}+\frac{9{\mathrm{k}}^2}{{\mathrm{b}}^2}=1$ or $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{9{\mathrm{y}}^2}{{\mathrm{b}}^2}=1(\because (\mathrm{h},\mathrm{k})$ are arbitrary $)$

8. If $\mathrm{C}$ is the centre of the ellipse $9{\mathrm{x}}^2+16{\mathrm{y}}^2=144$ and $\mathrm{S}$ is one focus. The ratio of $\mathrm{CS}$ to semi major axis is

(a) $\sqrt{7}:16$

(b) $\sqrt{7}:4$

(c) $\sqrt{5}:\sqrt{7}$

(d) None of these

Practice questions

1. The equation of the circle drawn with the two foci of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ as the end points of a diameter is

(a) $x^2+y^2=a^2+b^2$

(b) ${\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2$

(c) $x^2+y^2=2a^2$

(d) ${\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2-{\mathrm{b}}^2$

2. The radius of the circle passing through the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$ and having its centre $(0,3)$ is

(a) $3\sqrt{2}$

(b) $3$

(c) $\sqrt{12}$

(d) $\frac{7}{2}$

3. $\frac{{\mathrm{x}}^2}{{\mathrm{r}}^2-\mathrm{r}-6}+\frac{{\mathrm{y}}^2}{{\mathrm{r}}^2-6\mathrm{r}+5}=1$ will represents the ellipse, if $\mathrm{r}$ lies in the interval

(a) $(-\mathrm{\infty },2)$

(b) $(3,\mathrm{\infty })$

(c) $(5,\mathrm{\infty })$

(d) $(1,\mathrm{\infty })$

4. The semi latus rectum of an ellipse is

(a) The AM of the segments of its focal chord.

(b) The GM of the segments of its focal chord

(c) The HM of the segments of its focal chord

(d) None of these

5. The following equation represents an ellipse $25(x^2-6x+9)+16y^2=400$. How should the axes be transformed so that the ellipse is represented by the equation $\frac{x^2}{16}+\frac{y^2}{25}=1$______

6. Let $\mathrm{P}$ be a variable point on the ellipse $\frac{{\mathrm{x}}^2}{16}+\frac{{\mathrm{y}}^2}{25}=1$ with foci ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$. It $\mathrm{A}$ be area of the triangle ${\mathrm{PS}}_1{\mathrm{S}}_2$ then the maximum value of $\mathrm{A}$ is ________

7. In an ellipse, if the lines joining a focus to the extremities of the minor axis make an equilateral triangle with the minor axis, the eccentricity of the ellipse is

(a) $3/4$

(b) $\sqrt{3}/2$

(c) $1/2$

(d) $2/3$

8. Column Matching:

For the ellipse $\frac{x^2}{5}+\frac{y^2}{4}=1$

9. The centre of the ellipse $14x^2-4xy+11y^2-44x-58y+71=0$ is$\dots \dots ..$