# Conic Sections - Online Test

Q1. If the line 2x – y +  = 0 is a diameter of the circle  then  =
Answer : Option A
Explaination / Solution:

Equation of circle is

Applying completing the square method

Comparing the above equation with   we get center as (-3,3) and radius as    .

As centre of the circlre lies on diameter , it will satisfy the equation of diameter, so on putting (-3,3) in equation of diameter we get

=>

=>

Q2. Length of common chord of the circles and  is
Answer : Option D
Explaination / Solution:

Applying the completing the square

Comparing the above equation with  we get its center as (-1,-3) and radius (r1)=

Comparing the above equation with  we get its center as (2,1) and radius (r2)=

So distance(d) between the centers is   units
After applying general formula of length of common chord as  and putting the values of r1, r2 and d we get length as
.

Q3. The circles  and
Answer : Option B
Explaination / Solution:

Circle touches externally if distance between the centers is equal to the sum of the radii .

After applying completing the square, we get

so the center is (-3,-3) and radius is

After applying completing the square

so the center is (6,6) and radius is
Distance between centres  and sum of radii   are equal.

Hence the circle touches externally.

Q4.

The equation  represents

Answer : Option C
Explaination / Solution:

The above circle can be written as   Here the center is (0,0) and radius is also 0 units.

So it is a degenerate circle as  degenerate circle is a circle( a point) where radius is zero units.

Q5.

The equation represents

Answer : Option C
Explaination / Solution:

The general equation of the circle is x2+y2-2gh-2fy+c = 0. Sice the given equations satisfies the general equation, it represents the equation of the circle.

Q6. Circumcentre of the triangle, whose vertices are (0, 0), (6, 0) and (0, 4) is
Answer : Option A
Explaination / Solution:

circumcentre of a right angled triangle ABC right angled at A is  as circumcentre of right angled triangle lies on the mid pont of the hypotenuse.

so mid point of BC=(,) i.e.(3,2)

Q7. If (x-a)+ (y-b)= c2 represents a circle, then
Answer : Option D
Explaination / Solution:

(x-a)+ (y-b)= c2  here (a,b) is center and c is the radius

and radius cannot be zero because if radius is zero it will become a point or degenerate circle so c0.

Q8. Two perpendicular tangents to the circle  meet at P. The locus of P is
Answer : Option C
Explaination / Solution:

locus of P is a circle with centre at origin and radius .This is known as the director circle of the circle
Q9. The number of tangents to the circle which pass through the point ( 3, - 2), is
Answer : Option D
Explaination / Solution:

After completing the square,we get

so center is (4,3) and radius is 4.

Distance between center and given point   which is greater than 4.

hence point lies outside the circle .

Since point lies outside of the circle there will be 2 tangents since two tangents can be drawn from external point to a circle.

Q10. The length of the chord joining the point ( 4 cos , 4 sin ) and 4 ( cos(+), 4 sin( + )) of the circle  is
Answer : Option A
Explaination / Solution:

using distance formulae
on simplifying we get,