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?

  • 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?

    <p>Completing the square</p> Signup and view all the answers

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

    <p>x^2 + y^2 + Dx + Ey + F = 0</p> Signup and view all the answers

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

    <p>OP = sqrt((x - 0)^2 + (y - 0)^2)</p> Signup and view all the answers

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

    <p>x^2 + y^2 = r^2</p> Signup and view all the answers

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

    <p>(3, 4)</p> Signup and view all the answers

    What is the definition of a tangent to a circle?

    <p>A straight line that touches the circle at exactly one point without crossing it</p> Signup and view all the answers

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

    <p>The product of their gradients is -1</p> Signup and view all the answers

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

    <p>Write the equation of the circle in its standard form</p> Signup and view all the answers

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

    <p>To write the equation of the tangent</p> Signup and view all the answers

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

    <p>To find the gradient of the tangent</p> Signup and view all the answers

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

    <p>The center of the circle</p> 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?

    <p>To write the equation of the circle in standard form</p> Signup and view all the answers

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

    <p>The radius is perpendicular to the tangent at the point of tangency</p> Signup and view all the answers

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

    <p>To determine the center and radius of the circle</p> 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?

    <p>By calculating the slope of the line segment from the center to the point of tangency</p> Signup and view all the answers

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

    <p>Their product is -1</p> Signup and view all the answers

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

    <p>Find the gradient of the tangent</p> Signup and view all the answers

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

    <p>To write the equation of the tangent in gradient-point form</p> Signup and view all the answers

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

    <p>y - y1 = m(x - x1)</p> Signup and view all the answers

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

    <p>The point of tangency</p> Signup and view all the answers

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

    <p>Because of the definition of a tangent to a circle</p> 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?

    <p>x^2 + y^2 = r^2</p> Signup and view all the answers

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

    <p>The line y = 2x</p> 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?

    <p>(x - a)^2 + (y - b)^2 = r^2</p> Signup and view all the answers

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

    <p>To rewrite the equation in a more simplified form</p> 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?

    <p>Complete the square for both the x and y terms</p> 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?

    <p>It is the center of the circle</p> Signup and view all the answers

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

    <p>(x + 2)^2 + (y - 3)^2 = 16</p> Signup and view all the answers

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

    <p>They are perpendicular</p> 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?

    <p>Circle</p> 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?

    <p>4</p> 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?

    <p>To find the center and radius of the circle</p> 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?

    <p>(2, -3)</p> 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?

    <p>They are perpendicular</p> Signup and view all the answers

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

    <p>(x + 1)^2 + (y - 2)^2 = 9</p> 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?

    <p>It is the center of the circle</p> 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?

    <p>Complete the square for the x terms and the y terms</p> 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?

    <p>-D/2</p> Signup and view all the answers

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

    <p>m_radius = -1/m_tangent</p> Signup and view all the answers

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

    <p>To find the gradient of the tangent</p> 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?

    <p>(a, b)</p> Signup and view all the answers

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

    <p>y - y1 = m_tangent (x - x1)</p> Signup and view all the answers

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

    <p>The point of tangency is on the circumference of the circle</p> Signup and view all the answers

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

    <p>Because of the definition of a tangent</p> Signup and view all the answers

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

    <p>To find the center of the circle</p> Signup and view all the answers

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