Ellipse properties Quiz 1

Question 1

Which of the following ellipses has coordinates of foci $$\left( \pm \dfrac{2}{\sqrt{105}},0 \right)$$

Accepted Answer

$$15{{x}^{2}}+35{{y}^{2}}=1$$

Answer

$$35{{x}^{2}}+15{{y}^{2}}=1$$

Answer

$$6{{x}^{2}}+5{{y}^{2}}=1$$

Answer

$$19{{x}^{2}}+6{{y}^{2}}=1$$

Question 2

At which points on the $$x-$$axis, the ellipse $$17\left( {{x}^{2}}+3{{y}^{2}} \right)=9$$ intersects. Also, write the equation of the ellipse in standard form.

Accepted Answer

$$	ext{point}=\left( \pm \dfrac{3}{\sqrt{17}},0 ight),	ext{equation}=\dfrac{{{x}^{2}}}{\left( \dfrac{9}{17} ight)}+\dfrac{{{y}^{2}}}{\left( \dfrac{3}{17} ight)}=1$$

Answer

$$	ext{point}=\left( \pm \dfrac{4}{\sqrt{17}},0 ight),	ext{equation}=\dfrac{{{x}^{2}}}{\left( \dfrac{9}{17} ight)}+\dfrac{{{y}^{2}}}{\left( \dfrac{3}{17} ight)}=1$$

Answer

$$	ext{point}=\left( 0,\pm \dfrac{3}{\sqrt{17}} ight),	ext{equation}=\dfrac{{{x}^{2}}}{\left( \dfrac{3}{17} ight)}+\dfrac{{{y}^{2}}}{\left( \dfrac{9}{17} ight)}=1$$

Answer

$$	ext{point}=\left( 0,\pm \dfrac{2}{\sqrt{17}} ight),	ext{equation}=\dfrac{{{x}^{2}}}{\left( \dfrac{3}{17} ight)}+\dfrac{{{y}^{2}}}{\left( \dfrac{9}{17} ight)}=1$$

Question 3

Write the value of length of latus rectum of ellipse $$133{{x}^{2}}+152{{y}^{2}}=112.$$

Accepted Answer

$$\dfrac{7\sqrt{19}}{19}$$

Answer

$$\dfrac{19\sqrt{7}}{7}$$

Answer

$$\dfrac{8\sqrt{19}}{19}$$

Answer

$$\dfrac{18\sqrt{7}}{7}$$

Question 4

Write the eccentricity of the ellipse $$5{{x}^{2}}+9{{y}^{2}}=29.$$

Accepted Answer

$$\dfrac{2}{3}$$

Answer

$$\dfrac{4}{5}$$

Answer

$$\dfrac{1}{9}$$

Answer

$$\dfrac{6}{7}$$

Question 5

Write the points on the ellipse $$6{{x}^{2}}+7{{y}^{2}}=1$$ where the tangent is parallel to $$x-$$axis.

Accepted Answer

$$\left( 0,\pm \dfrac{1}{\sqrt{7}} \right)$$

Answer

$$\left( 0,\pm \dfrac{1}{\sqrt{5}} \right)$$

Answer

$$\left( 0,\pm \dfrac{1}{\sqrt{3}} \right)$$

Answer

$$\left( 0,\pm \dfrac{1}{\sqrt{11}} \right)$$