Conic Sections Test

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Conic Sections Test - 49

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Weekly Quiz Competition

Question 1
1 / -0

The eccentricity of the conic $$9x^{2} + 25y^{2} = 225$$ is

A
$$\dfrac 25$$

B
$$\dfrac 45$$

C
$$\dfrac 13$$

D
$$\dfrac 15$$

E
$$\dfrac 35$$

Solution

Equation of the given ellipse is

$$9x^2+25y^2=225$$

Dividing through out by $$225$$

$$\dfrac {x^2}{25}+\dfrac {y^2}{9}=1$$

Hence $$a^2=25$$ and $$b^2=9$$

$$\therefore \ c=\sqrt {25-9}=\sqrt {16}=4$$

eccentricity $$(e)=\dfrac {c}{a}=\dfrac {4}{5}$$
Question 2
1 / -0

The eccentricity of the conic $$9{x}^{2}-16{y}^{2}=144$$ is

A
$$\cfrac{5}{4}$$

B
$$\cfrac{4}{3}$$

C
$$\cfrac{4}{5}$$

D
$$\sqrt{7}$$

Solution
Question 3
1 / -0

The vertex of the parabola $$y^2 - 4y - x + 3 = 0$$ is

A
$$(-1,3)$$

B
$$(-1,2)$$

C
$$(2,-1)$$

D
$$(3, -1)$$

Solution

We have,

$$y^2 - 4y - x + 3 = 0$$
$$\Rightarrow (y-2)^2 - 4 - x+3 = 0$$

$$\Rightarrow (y-2)^2 = (x+1)$$

$$\therefore$$ Vertex of the parabola $$= (-1,2)$$
Question 4
1 / -0

The equation $$5x^2 + y^2 + y = 8$$ represents

A
An ellipse

B
A parabola

C
A hyperbola

D
A Circle

Solution

We have,
$$5x^2 + y^2 + y = 8$$
$$\Rightarrow 5x^2 + \left(y + \dfrac{1}{2}\right)^2 - \left(\dfrac{1}{2}\right)^2 = 8$$

$$\Rightarrow 5x^2 + \left(y + \dfrac{1}{2}\right)^2 = \dfrac{33}{8}$$

$$\Rightarrow \dfrac{\dfrac{5x^2}{33}}{8} + \dfrac{\dfrac{\left(y+\dfrac{1}{2}\right)^2}{33}}{8} = 1$$

$$\Rightarrow \dfrac{x^2}{\left(\dfrac{33}{40}\right)} + \dfrac{\left(y+\dfrac{1}{2}\right)^2}{\left(\dfrac{33}{8}\right)} = 1$$
which is sn equation of ellipse.
Question 5
1 / -0

Write the length of the latus-rectum of the hyperbola $$16{x}^{2}-9{y}^{2}=144$$

A
$$\cfrac{5}{3}$$

B
$$\cfrac{4}{3}$$

C
$$\cfrac{3}{4}$$

D
$$\cfrac{4}{5}$$

Solution
Question 6
1 / -0

The equation of the circle with centre $$(2, 2)$$ which passes through $$(4,5)$$ is

A
$$x^2 + y^2 - 4x + 4y - 77 = 0$$

B
$$x^2 + y^2 - 4x - 4y - 5 = 0$$

C
$$x^2 + y^2 + 2x + 2y - 59 = 0$$

D
$$x^2 + y^2 - 2x - 2y - 23 = 0$$

E
$$x^2 + y^2 + 4x - 2y -26 = 0$$

Solution

Radius of circle is $$\sqrt{(4-2)^2+(5-2)^2}=\sqrt{13}$$ So, equation of circle is
$$(x-2)^2+(y-2)^2=13$$
$$\Rightarrow x^2+4-4x+y^2+4-4y=13$$
$$\Rightarrow x^2+y^2-4x-4y-5=0$$
Question 7
1 / -0

The centre of the ellipse $$4x^2 + y^2 - 8x + 4y - 8 = 0$$ is

A
$$(0,2)$$

B
$$(2,-1)$$

C
$$(2,1)$$

D
$$(1,2)$$

Solution

We have,

$$4x^2 + y^2 - 8x + 4y - 8 = 0$$
$$\Rightarrow (4x^2 - 8x) + (y^2 + 4y)-8=0$$
$$\Rightarrow (4x^2 - 2x) + (y^2 + 4y)-8=0$$

$$\Rightarrow 4[(x-1)^2 - 1]+[(y + 2)^2 - 4]-8=0$$
$$\Rightarrow 4(x-1)^2 -4 + (y + 2)^2 - 4 -8=0$$

$$\Rightarrow 4(x-1)^2 + (y+2)^2 = 16$$

$$\Rightarrow \dfrac{(x-1)^2}{4} + \dfrac{(y+2)^2}{16} = 1$$

$$\therefore$$ Centre $$= (1,2)$$
Question 8
1 / -0

The latus-rectum of the hyperbola $$16{x}^{4}-9{y}^{2}=144$$ is

A
$$16/3$$

B
$$32/3$$

C
$$8/3$$

D
$$4/3$$

Solution
Question 9
1 / -0

Write the eccentricity of the hyperbola whose latus-rectum is half of its transverse axis.

A
$$\cfrac{1}{\sqrt 3}$$

B
$$\cfrac{1}{\sqrt 5}$$

C
$$\cfrac{1}{\sqrt 2}$$

D
None of the above

Solution
Question 10
1 / -0

The eccentricity of the ellipse $$\dfrac{(x-1)^2}{2} + \left(y + \dfrac{3}{4}\right)^2 = \dfrac{1}{16}$$ is

A
$$\dfrac{1}{\sqrt{2}}$$

B
$$\dfrac{1}{2\sqrt{2}}$$

C
$$\dfrac{1}{2}$$

D
$$\dfrac{1}{4}$$

Solution

$$\dfrac{(x-1)^2}{2} + \left(y + \dfrac{3}{4}\right)^2 = \dfrac{1}{16}$$

$$\Rightarrow 8(x-1)^2 + 16\left(y +\dfrac{3}{4}\right)^2 = 1$$

$$\Rightarrow \dfrac{(x-1)^2}{\dfrac{1}{8}} + \dfrac{\left(y + \dfrac{3}{4}\right)^2}{\dfrac{1}{16}} = 1$$

$$\therefore a^2 = \dfrac{1}{8}$$ and $$b^2 = \dfrac{1}{16}$$

$$\Rightarrow a = \dfrac{1}{2\sqrt{2}}$$ and $$b = \dfrac{1}{4}$$

$$\therefore e = \sqrt{1 - \dfrac{b^2}{a^2}}$$ $$[\because a > b]$$

$$= \sqrt{1-\dfrac{\dfrac{1}{16}}{\dfrac{1}{8}}} = \sqrt{1 - \dfrac{1}{2}} = \dfrac{1}{\sqrt{2}}$$

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Re-Attempt Weekly Quiz Competition

Conic Sections Test - 49

Result

Conic Sections Test - 49

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SHARING IS CARING

Question 1

$$\dfrac 25$$

$$\dfrac 45$$

$$\dfrac 13$$

$$\dfrac 15$$

$$\dfrac 35$$

Solution

Question 2

$$\cfrac{5}{4}$$

$$\cfrac{4}{3}$$

$$\cfrac{4}{5}$$

$$\sqrt{7}$$

Solution

Question 3

$$(-1,3)$$

$$(-1,2)$$

$$(2,-1)$$

$$(3, -1)$$

Solution

Question 4

An ellipse

A parabola

A hyperbola

A Circle

Solution

Question 5

$$\cfrac{5}{3}$$

$$\cfrac{4}{3}$$

$$\cfrac{3}{4}$$

$$\cfrac{4}{5}$$

Solution

Question 6

$$x^2 + y^2 - 4x + 4y - 77 = 0$$

$$x^2 + y^2 - 4x - 4y - 5 = 0$$

$$x^2 + y^2 + 2x + 2y - 59 = 0$$

$$x^2 + y^2 - 2x - 2y - 23 = 0$$

$$x^2 + y^2 + 4x - 2y -26 = 0$$

Solution

Question 7

$$(0,2)$$

$$(2,-1)$$

$$(2,1)$$

$$(1,2)$$

Solution

Question 8

$$16/3$$

$$32/3$$

$$8/3$$

$$4/3$$

Solution

Question 9

$$\cfrac{1}{\sqrt 3}$$

$$\cfrac{1}{\sqrt 5}$$

$$\cfrac{1}{\sqrt 2}$$

None of the above

Solution

Question 10

$$\dfrac{1}{\sqrt{2}}$$

$$\dfrac{1}{2\sqrt{2}}$$

$$\dfrac{1}{2}$$

$$\dfrac{1}{4}$$

Solution

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