Gr12 Mathematics: Ch 6 Sum Analytical Geometry

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What is the equation of a circle with center at the origin and radius r?

x^2 - y^2 = r^2
x^2 - y^2 = -r^2
x^2 + y^2 = -r^2
x^2 + y^2 = r^2 (correct)

What is the symmetry of a circle with center at the origin?

Only about the x-axis
Only about the y-axis
About the x-axis, y-axis, origin, and lines y = x and y = -x (correct)
None of the above

What is the equation of a circle with center at (a, b) and radius r?

(x + a)^2 + (y + b)^2 = r^2
(x - a)^2 + (y + b)^2 = r^2
(x - a)^2 + (y - b)^2 = r^2 (correct)
(x - a)^2 - (y - b)^2 = r^2

How do you find the center and radius of a circle given its general form equation?

Completing the square (C) Signup and view all the answers

What is the general form of a circle's equation?

x^2 + y^2 + Dx + Ey + F = 0 (D) Signup and view all the answers

What is the distance formula used to derive the equation of a circle?

OP = sqrt((x - 0)^2 + (y - 0)^2) (B) Signup and view all the answers

What is the Pythagorean theorem used to derive in the equation of a circle?

x^2 + y^2 = r^2 (A) Signup and view all the answers

What is the center of a circle with equation (x - 3)^2 + (y - 4)^2 = 9?

(3, 4) (A) Signup and view all the answers

What is the definition of a tangent to a circle?

A straight line that touches the circle at exactly one point without crossing it (B) Signup and view all the answers

What is the relationship between the gradients of the radius and the tangent at the point of tangency?

The product of their gradients is -1 (D) Signup and view all the answers

What is the first step in determining the equation of a tangent to a circle?

Write the equation of the circle in its standard form (C) Signup and view all the answers

What is the gradient-point form of the straight line equation used for?

To write the equation of the tangent (C) Signup and view all the answers

What is the purpose of finding the gradient of the radius?

To find the gradient of the tangent (B) Signup and view all the answers

What is the meaning of the coordinates (a, b) in the standard form of the circle equation?

The center of the circle (C) Signup and view all the answers

What is the purpose of completing the square for the x terms and the y terms in the equation of the circle?

To write the equation of the circle in standard form (A) Signup and view all the answers

What is the relationship between the point of tangency and the radius of the circle?

The radius is perpendicular to the tangent at the point of tangency (D) Signup and view all the answers

What is the purpose of rewriting the equation of the circle in its standard form?

To determine the center and radius of the circle (D) Signup and view all the answers

Given the equation of a circle, how do you find the gradient of the radius at the point of tangency?

By calculating the slope of the line segment from the center to the point of tangency (C) Signup and view all the answers

What is the relationship between the gradients of the radius and the tangent at the point of tangency?

Their product is -1 (A) Signup and view all the answers

What is the next step after finding the gradient of the radius at the point of tangency?

Find the gradient of the tangent (C) Signup and view all the answers

What is the purpose of finding the gradient of the tangent?

To write the equation of the tangent in gradient-point form (A) Signup and view all the answers

What is the equation of the tangent in gradient-point form?

y - y1 = m(x - x1) (B) Signup and view all the answers

What is the point (x1, y1) in the equation of the tangent?

The point of tangency (A) Signup and view all the answers

Why is the tangent to a circle perpendicular to the radius at the point of tangency?

Because of the definition of a tangent to a circle (B) Signup and view all the answers

What is the correct formula to derive the equation of a circle with center at the origin and radius r?

x^2 + y^2 = r^2 (A) Signup and view all the answers

Which of the following is NOT a symmetry of a circle with center at the origin?

The line y = 2x (D) Signup and view all the answers

What is the equation of a circle with center at (a, b) and radius r if the point (a + r, b) lies on the circle?

(x - a)^2 + (y - b)^2 = r^2 (C) Signup and view all the answers

What is the purpose of completing the square in rewriting a circle's equation in standard form?

To rewrite the equation in a more simplified form (A) Signup and view all the answers

If the equation of a circle is x^2 + y^2 - 6x + 8y - 15 = 0, what is the correct step to find the center and radius of the circle?

Complete the square for both the x and y terms (C) Signup and view all the answers

What is the significance of the point (a, b) in the equation of a circle with center at (a, b) and radius r?

It is the center of the circle (D) Signup and view all the answers

What is the correct equation of a circle with center at (-2, 3) and radius 4?

(x + 2)^2 + (y - 3)^2 = 16 (C) Signup and view all the answers

What is the relationship between the derivatives of the radius and the tangent at the point of tangency?

They are perpendicular (D) Signup and view all the answers

What is the geometric shape formed by the set of all points in a plane that are equidistant from a fixed point called the center?

Circle (C) Signup and view all the answers

If a circle with center at the origin has the equation x^2 + y^2 = 16, what is the radius of the circle?

4 (B) Signup and view all the answers

What is the purpose of completing the square when rewriting the equation of a circle from its general form to its standard form?

To find the center and radius of the circle (D) Signup and view all the answers

If a circle has the equation (x - 2)^2 + (y + 3)^2 = 25, what are the coordinates of its center?

(2, -3) (A) Signup and view all the answers

What is the relationship between the gradients of the radius and the tangent at the point of tangency on a circle?

They are perpendicular (A) Signup and view all the answers

What is the equation of a circle with center at (-1, 2) and radius 3?

(x + 1)^2 + (y - 2)^2 = 9 (C) Signup and view all the answers

What is the significance of the point (a, b) in the equation of a circle with center at (a, b) and radius r?

It is the center of the circle (B) Signup and view all the answers

If a circle has the equation x^2 + y^2 - 8x + 12y - 20 = 0, what is the first step in finding the center and radius of the circle?

Complete the square for the x terms and the y terms (C) Signup and view all the answers

If the equation of a circle is given as x^2 + Dx + y^2 + Ey = -F, what is the value of a in the standard form of the equation?

-D/2 (C) Signup and view all the answers

What is the relationship between the gradients of the radius and the tangent at the point of tangency?

m_radius = -1/m_tangent (B) Signup and view all the answers

What is the purpose of finding the gradient of the radius at the point of tangency?

To find the gradient of the tangent (D) Signup and view all the answers

If the equation of a circle is given as (x - a)^2 + (y - b)^2 = r^2, what is the coordinate of the center of the circle?

(a, b) (A) Signup and view all the answers

What is the equation of the tangent to a circle at the point of tangency (x1, y1)?

y - y1 = m_tangent (x - x1) (B) Signup and view all the answers

What is the relationship between the point of tangency and the radius of the circle?

The point of tangency is on the circumference of the circle (B) Signup and view all the answers

Why is the tangent to a circle perpendicular to the radius at the point of tangency?

Because of the definition of a tangent (D) Signup and view all the answers

What is the purpose of rewriting the equation of the circle in its standard form?

To find the center of the circle (C) Signup and view all the answers

Gr12 Mathematics: Ch 6 Sum Analytical Geometry

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Questions and Answers

What is the equation of a circle with center at the origin and radius r?

What is the symmetry of a circle with center at the origin?

What is the equation of a circle with center at (a, b) and radius r?

How do you find the center and radius of a circle given its general form equation?

What is the general form of a circle's equation?

What is the distance formula used to derive the equation of a circle?

What is the Pythagorean theorem used to derive in the equation of a circle?

What is the center of a circle with equation (x - 3)^2 + (y - 4)^2 = 9?

What is the definition of a tangent to a circle?

What is the relationship between the gradients of the radius and the tangent at the point of tangency?

What is the first step in determining the equation of a tangent to a circle?

What is the gradient-point form of the straight line equation used for?

What is the purpose of finding the gradient of the radius?

What is the meaning of the coordinates (a, b) in the standard form of the circle equation?

What is the purpose of completing the square for the x terms and the y terms in the equation of the circle?

What is the relationship between the point of tangency and the radius of the circle?

What is the purpose of rewriting the equation of the circle in its standard form?

Given the equation of a circle, how do you find the gradient of the radius at the point of tangency?

What is the relationship between the gradients of the radius and the tangent at the point of tangency?

What is the next step after finding the gradient of the radius at the point of tangency?

What is the purpose of finding the gradient of the tangent?

What is the equation of the tangent in gradient-point form?

What is the point (x1, y1) in the equation of the tangent?

Why is the tangent to a circle perpendicular to the radius at the point of tangency?

What is the correct formula to derive the equation of a circle with center at the origin and radius r?

Which of the following is NOT a symmetry of a circle with center at the origin?

What is the equation of a circle with center at (a, b) and radius r if the point (a + r, b) lies on the circle?

What is the purpose of completing the square in rewriting a circle's equation in standard form?

If the equation of a circle is x^2 + y^2 - 6x + 8y - 15 = 0, what is the correct step to find the center and radius of the circle?

What is the significance of the point (a, b) in the equation of a circle with center at (a, b) and radius r?

What is the correct equation of a circle with center at (-2, 3) and radius 4?

What is the relationship between the derivatives of the radius and the tangent at the point of tangency?

What is the geometric shape formed by the set of all points in a plane that are equidistant from a fixed point called the center?

If a circle with center at the origin has the equation x^2 + y^2 = 16, what is the radius of the circle?

What is the purpose of completing the square when rewriting the equation of a circle from its general form to its standard form?

If a circle has the equation (x - 2)^2 + (y + 3)^2 = 25, what are the coordinates of its center?

What is the relationship between the gradients of the radius and the tangent at the point of tangency on a circle?

What is the equation of a circle with center at (-1, 2) and radius 3?

What is the significance of the point (a, b) in the equation of a circle with center at (a, b) and radius r?

If a circle has the equation x^2 + y^2 - 8x + 12y - 20 = 0, what is the first step in finding the center and radius of the circle?

If the equation of a circle is given as x^2 + Dx + y^2 + Ey = -F, what is the value of a in the standard form of the equation?

What is the relationship between the gradients of the radius and the tangent at the point of tangency?

What is the purpose of finding the gradient of the radius at the point of tangency?

If the equation of a circle is given as (x - a)^2 + (y - b)^2 = r^2, what is the coordinate of the center of the circle?

What is the equation of the tangent to a circle at the point of tangency (x1, y1)?

What is the relationship between the point of tangency and the radius of the circle?

Why is the tangent to a circle perpendicular to the radius at the point of tangency?

What is the purpose of rewriting the equation of the circle in its standard form?

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