JEE Main Mathematics: Co-ordinate Geometry Quiz!

1. If the two circles x² + y² = r² and x² + y² – 10x + 16 = 0 intersect each other at some real and distinct points then which of the following is true?

2 < r < 8

R = 2

R < 3

R > 5

Explanation

2. Lines x² – 8x + 12 = 0 and y² – 14y + 45 = 0 form a square. What will be the center of the circle inscribed in the square?

(4, 3)

(3, 4)

(4, 7)

(7, 4)

Explanation

3. If a₁, a₂, a₃ as well as b₁, b₂, b₃ are in G.P. with the same common ratio, then the points (a₁, b₁), (a₂, b₂) and (a₃, b₃):

Are vertices of a triangle

Lie on parabola

Lie on a circle

Lie on a straight line

Given, all the three points are related to each other in a way that their common ratios are equal. On calculating the area of the shape formed by these three points, area is found to be zero which means that these three points are collinear.

Explanation

Given, all the three points are related to each other in a way that their common ratios are equal. On calculating the area of the shape formed by these three points, area is found to be zero which means that these three points are collinear.

4. The angle between the pair of straight lines y²sin²θ – xysin²θ + x² (cos²θ – 1) = 0 is:

Explanation

5. What will be the equation of the straight line passing from the point (–3, 5) and making intercept of length

fraction numerator 2 over denominator square root of 10 end fraction

between the lines x + 3y +1 = 0 and x + 3y – 1 = 0?

X + 2 = 0

3x – y + 13 = 0

3x + 4y =18

4x + 3y = 12

The given lines are parallel and the distance between them is 2/√10 which the perpendicular distance between the two is given parallel lines. So the line cutting the given segment must be perpendicular to the x + 3y + 1 = 0 and pass through the point (–3, 5).
Equation of the line is 3(x + 3) – (y – 4) = 0; 3x – y + 13 = 0.

Explanation

The given lines are parallel and the distance between them is 2/√10 which the perpendicular distance between the two is given parallel lines. So the line cutting the given segment must be perpendicular to the x + 3y + 1 = 0 and pass through the point (–3, 5).
Equation of the line is 3(x + 3) – (y – 4) = 0; 3x – y + 13 = 0.

6. If a, b, c are related by 9a²+ 16b² – 16c² + 24ab = 0then the greatest distance between any two lines of the family of lines ax + by + c = 0 is:

4/3

Explanation

7. Circles x² + y² + 2ax + c = 0 and circle x² +y² +2by + c = 0, touches each other then which of the following is correct?

1/a + 1/b = 1/c

The given equation represents two circles. If the circles touch each other, it means they have a common tangent at the point of contact. The equation of a tangent to a circle is given by the form ax + by + c = 0. Comparing the equations of the circles with the tangent equation, we can see that the coefficients of x and y are the same. Therefore, we can equate the coefficients of x and y in the two circle equations. By doing so, we get 1/a + 1/b = 1/c.

Explanation

The given equation represents two circles. If the circles touch each other, it means they have a common tangent at the point of contact. The equation of a tangent to a circle is given by the form ax + by + c = 0. Comparing the equations of the circles with the tangent equation, we can see that the coefficients of x and y are the same. Therefore, we can equate the coefficients of x and y in the two circle equations. By doing so, we get 1/a + 1/b = 1/c.

8. The line y = x meets y = k

e to the power of x

at: (Given k ≤ 0)

One point

Two points

Three points

Zero points

Explanation

9. The equation: x²y² – 2xy² – 3y² – 4x²y + 8xy + 12y = 0; represents:

A pair of lines

A circle and pair of lines

Four lines

None of these

Explanation

10. The number of common tangents to the circle x² + y² – 4 = 0 and x² + y² – 6x – 8y – 24 = 0 is:

0

1

3

∞

From the two equations C1C2 = 3√10; which is equal to the sum of the two radius of the given circle, it means that the two circles touch each other externally which means that both the circles will have one common tangent and as they are touching each other externally then there will be two more tangents which will touch these

Explanation

From the two equations C1C2 = 3√10; which is equal to the sum of the two radius of the given circle, it means that the two circles touch each other externally which means that both the circles will have one common tangent and as they are touching each other externally then there will be two more tangents which will touch these