Points of intersection of a line and a circle - One Line Questions
1.
Find the points of intersection of the line x = -1 and the circle x^2 + y^2 = 1. —
(-1, 0)
2.
Find the points of intersection of the line y = 5 and the circle x^2 + y^2 = 25. —
(0, 5)
3.
Find the points of intersection of the line x + y = 1 and the circle x^2 + y^2 = 1. —
(1, 0) and (0, 1)
4.
Find the points of intersection of the line x - y = 0 and the circle x^2 + y^2 = 2. —
(1, 1) and (-1, -1)
5.
Find the points of intersection of the line x + y = 2 and the circle x^2 + y^2 = 2. —
(1, 1)
6.
Consider the circle x^2 + y^2 = 5 and the line y = 2x. The points of intersection are: —
(1, 2) and (-1, -2)
7.
Consider the circle x^2 + y^2 - 4x - 6y + 9 = 0 and the line x = 1. Find the points of intersection. —
(1, 3 + sqrt(7)) and (1, 3 - sqrt(7))
8.
Find the points of intersection of the line y = x and the circle x^2 + y^2 = 8. —
(2, 2) and (-2, -2)
9.
Consider the circle x^2 + y^2 = 13 and the line y = x + 1. Substituting y gives x^2 + (x+1)^2 = 13, which simplifies to 2x^2 + 2x - 12 = 0, or x^2 + x - 6 = 0. The roots are x = 2 and x = -3. The corresponding y values are: —
(2, 3) and (-3, -2)
10.
Find the points of intersection of the line x = 3 and the circle x^2 + y^2 = 25. —
(3, 4) and (3, -4)
11.
Find the points of intersection of the line 3x + 4y = 25 and the circle x^2 + y^2 = 25. —
(3, 4) only
12.
The circle x^2 + y^2 = 16. The line x = 4. The point of intersection is: —
(4, 0)
13.
The equation of a circle is x^2 + y^2 - 2x - 4y - 4 = 0. The equation of a line is x = 4. Find the points of intersection. —
(4, 2 + sqrt(8)) and (4, 2 - sqrt(8))
14.
The line x + y = 10 is tangent to the circle x^2 + y^2 = 50. What is the point of tangency? —
(5, 5)
15.
Find the number of points of intersection between the circle x^2 + y^2 = 4 and the line x + y = 3. —
1
16.
The equation of the circle is x^2 + y^2 = 16. The equation of the line is y = 2x + c. For the line to be tangent, the value of c^2 must be: —
80
17.
For the circle x^2 + y^2 = r^2, the line x = r is: —
A tangent at (r, 0)
18.
If the discriminant of the quadratic equation obtained from the intersection of a line and a circle is zero, the line is: —
A tangent
19.
The circle x^2 + y^2 = 100 and the line y = mx + c. If c^2 = 100(1+m^2), then the line is: —
A tangent
20.
The circle x^2 + y^2 = 4. The line x + y = 2*sqrt(2). This line is: —
A tangent
21.
What is the general equation of a circle? —
x^2 + y^2 + 2gx + 2fy + c = 0
22.
What is the condition for the line y = c to be tangent to the circle x^2 + y^2 = r^2? —
c^2 = r^2
23.
What is the condition for the line y = mx + c to intersect the circle x^2 + y^2 = r^2 at two distinct points? —
c^2 < r^2(1 + m^2)
24.
What is the condition for the line ax + by + c = 0 to be tangent to the circle x^2 + y^2 = r^2? —
c^2 = r^2(a^2 + b^2)
25.
The line ax + by + c = 0 intersects the circle (x-h)^2 + (y-k)^2 = r^2. If the substitution leads to a quadratic equation with discriminant D, then: —
D > 0 implies two points, D = 0 implies one point, D < 0 implies no points
26.
The equation of a circle is (x-h)^2 + (y-k)^2 = r^2 and the equation of a line is ax + by + c = 0. What is the condition for the line to be tangent to the circle? —
Distance from (h, k) to the line is equal to r
27.
If a line passes through the center of a circle, the distance from the center to the line is: —
Zero
28.
If the quadratic equation resulting from the intersection of a line and a circle has no real roots (two complex conjugate roots), the line: —
Does not intersect the circle
29.
If the distance from the center of a circle to a line is greater than its radius, the line: —
Does not intersect the circle
30.
If the distance from the center of a circle to a line is equal to its radius, the line: —
Is a tangent
31.
If the distance from the center of a circle to a line is less than its radius, the line: —
Is a secant
32.
The equation x^2 + y^2 = 25. If we substitute y = k in this equation, and the resulting equation in x has no real solution, what does it imply about the line y = k? —
It does not intersect the circle
33.
What is the condition for the line lx + my + n = 0 to be a tangent to the circle x^2 + y^2 = r^2? —
n^2 = r^2(l^2 + m^2)
34.
If the line passes through the center of the circle, it will intersect the circle at: —
Two points (endpoints of a diameter)
35.
The equation of a circle is x^2 + y^2 = 9. The line y = 3 intersects the circle at: —
The line is a tangent at (0, 3)
36.
For the circle x^2 + y^2 = a^2 and the line x cos(alpha) + y sin(alpha) = p, the condition for tangency is: —
p = |a|
37.
For the circle x^2 + y^2 = r^2, the line y = mx + c intersects it at two distinct points if: —
r^2(1+m^2) > c^2
38.
If the quadratic equation resulting from the intersection of a line and a circle has exactly one real root (a repeated root), the line is called a: —
Tangent
39.
Consider the circle with center (h, k) and radius r. The line ax + by + c = 0 intersects the circle. The number of intersection points depends on the relationship between: —
The distance from (h, k) to the line and r
40.
What is the geometric interpretation of the condition r^2(1+m^2) = c^2 for the line y = mx + c and circle x^2 + y^2 = r^2? —
The distance from the origin to the line is equal to the radius.
41.
A line intersects a circle. If the resulting quadratic equation for the coordinates has complex roots, it means: —
The line does not intersect the circle
42.
To find the points of intersection of a line and a circle, we generally solve their equations simultaneously. —
True
43.
Consider the circle x^2 + y^2 - 2gx - 2fy + c = 0 and the line x = k. The intersection points are found by substituting x = k into the circle equation, leading to a quadratic in y. —
True
44.
The general equation of a circle passing through the intersection of the circle S = 0 and the line L = 0 is given by S + lambda*L = 0. —
False
45.
The general equation of a circle passing through the intersection of the circle S = 0 and the line L = 0 is given by S + lambda*L^2 = 0. —
True
46.
The circle x^2 + y^2 = 1 and the line x + y = sqrt(2). What is the nature of their intersection? —
One point (tangent)
47.
Consider the circle x^2 + y^2 = 25 and the line y = x + 1. Substituting y in the circle equation gives x^2 + (x+1)^2 = 25. What is the nature of the roots of the resulting quadratic equation? —
Two distinct real roots
48.
What is the general equation of a straight line? —
ax + by + c = 0
49.
If the line x = a intersects the circle x^2 + y^2 = r^2, the y-coordinates of the intersection points are given by: —
y = +/- sqrt(r^2 - a^2)
50.
If the system of equations for a line and a circle yields a quadratic equation with two distinct real roots, how many points of intersection are there? —
Two