Conic Sections Test

Result

Conic Sections Test - 40

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Weekly Quiz Competition

Question 1
1 / -0

The equation of the circle in the first quadrature touching each coordinate axis at a distance of one unit from the origin

A
$$x^{2}+y^{2}-2x-2y+1=0$$

B
$$x^{2}+y^{2}-2x-1=0$$

C
$$x^{2}+y^{2}-2x=0$$

D
None of these

Solution

We can clearly see from diagram that co-ordinates of centre of circle $$=(1,1)$$ & radius is unit
Thus equation of circle:-
$$(x-1)^{2}+(y-1)^{2}=1^{2}$$
$$x^{2}+y^{2}-2x-2y+1=0$$
Question 2
1 / -0

The equation of the circle which circumscribes the triangle formed by the lines x + y + 3 = 0, x - y + 1 =0 and x = 3 is

A
$$x^2+y^2-6x+2y-15=0$$

B
$$3x^2+3y^2-9=0$$

C
$$x^2+y^2+6x-2y+15=0$$

D
None of these

Solution
Question 3
1 / -0

The parametric form of the equation of the circle $${ x }^{ 2 }+{ y }^{ 2 }=9$$ is:

A
$$x=\sqrt { 3 } \cos { \theta } ,$$ $$y=\sqrt { 3 } \sin { \theta } $$

B
$$x=3\cos { \theta } ,\quad y=3\sin { \theta } $$

C
$$x=-\sqrt { 3 } \cos { \theta } ,$$ $$y=-\sqrt { 3 } \sin { \theta } $$

D
none of these

Solution
Question 4
1 / -0

An equilateral triangle is inscribed in the parabola $$y^2=4ax$$ with one vertex at the origin. The radius of the circumcircle of the triangle is

A
$$\dfrac{a}{2}$$

B
$$\dfrac{3a}{2}$$

C
$$4a$$

D
$$8a$$

Solution
Question 5
1 / -0

The equation of a circle with centre at $$(1, -2)$$ and passing through the centre of the given circle $$x^2+y^2+2y-3=0$$, is?

A
$$x^2+y^2-2x+4y+3=0$$

B
$$x^2+y^2-2x+4y-3=0$$

C
$$x^2+y^2+2x-4y-3=0$$

D
$$x^2+y^2+2x-4y+3=0$$

Solution
Question 6
1 / -0

The equation of circle whose centre is $$(1, -3)$$ and which touches the line $$2x-y-4=0$$, is

A
$$5x^2+5y^2+10x+30y+49=0$$

B
$$5x^2+5y^2+10x-30y-49=0$$

C
$$5x^2+5y^2-10x+30y-49=0$$

D
None of these

Solution
Question 7
1 / -0

The latus rectum of a parabola whose focal chord is $$PSQ$$ such that $$SP=3$$ and $$SQ=2$$, is given by

A
$$\dfrac{24}{5}$$

B
$$\dfrac{12}{5}$$

C
$$\dfrac{6}{5}$$

D
$$\dfrac{48}{5}$$

Solution

Given, $$SP=3$$
$$SQ=2$$
By property of parabola
The harmonic mean of the lengths two parts into which the focus divides the focal chord is equal to length of semi-latus rectum.
Let length of lotus rectum be L
$$\dfrac{2SP\times SQ}{SP+SQ}=\dfrac{1}{2}$$
$$L=\dfrac{4\times SP\times SQ}{SP+SQ}=\dfrac{4\times 3\times 2}{3+2}=\dfrac{24}{5}$$.
Question 8
1 / -0

The length of the latus rectum of the parabola $${ 2y }^{ 2 }+3y+4x-2=0$$ is ________.

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

B
$$\dfrac{ 1 }{ 3 } $$

C
$$2$$

D
None of these.

Solution
Question 9
1 / -0

The angle between the curves $$x^3-3xy^2=2$$ and $$3x^2y-y^3=2$$ is?

A
$$\dfrac{\pi}{6}$$

B
$$\dfrac{\pi}{4}$$

C
$$\dfrac{\pi}{3}$$

D
$$\dfrac{\pi}{2}$$

Solution
Question 10
1 / -0

Equation of the ellipse whose minor axis is equal to the distance between foci and whose latus rectum is $$10 ,$$ is given by ____________.

A
$$2 x ^ { 2 } + 3 y ^ { 2 } = 100$$

B
$$2 x ^ { 2 } + 3 y ^ { 2 } = 80$$

C
$$x ^ { 2 } + 2 y ^ { 2 } = 100$$

D
none of these

Solution