Circles
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Questions and Answers

If the distance between the centres of two circles is equal to the sum of their radii, how many direct common tangents can be drawn?

  • Three
  • Two (correct)
  • Four
  • One

    • What is the condition for two circles to intersect orthogonally?

  • |C1C2| = |r1 - r2| (correct)
  • r1 = r2
  • |C1C2| = r1 + r2
  • |C1C2| = √(r1² + r2²)

    • If the distance between the centres of two circles is less than the difference of their radii, what can be said about the tangents?

  • Two tangents are real and coincident (correct)
  • Two tangents are real and distinct
  • All four tangents are real
  • All four tangents are imaginary

    • What is the condition for two circles to have four imaginary common tangents?

    <p>|C1C2| &lt; |r1 - r2|</p> Signup and view all the answers

    If two circles have two direct common tangents that are real and distinct, what can be said about the distance between their centres?

    <p>It is less than the sum of their radii</p> Signup and view all the answers

    What is the equation of the circle with centre (h, k) and radius r?

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

    What is the equation of the circle drawn on the straight line joining (x1, y1) and (x2, y2) as diameter?

    <p>(x - x1)(x - x2) + (y - y1)(y - y2) = 0</p> Signup and view all the answers

    What is the condition for two circles to be concentric?

    <p>They have the same centre but different radii.</p> Signup and view all the answers

    What is the equation of the circle having centre at the origin and radius r?

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

    What is the equation of the circle in general form?

    <p>x^2 + y^2 + 2gx + 2fy + c = 0</p> Signup and view all the answers

    What is the point of intersection of the tangents at the point P (α) and Q (β) on the circle x^2 + y^2 = a^2?

    <p>(a cos(α + β)/2, a sin(α + β)/2)</p> Signup and view all the answers

    What is the equation of the straight line joining two points α and β on the circle x^2 + y^2 = a^2?

    <p>x cos(α + β) + y sin(α + β) = a cos(α + β)</p> Signup and view all the answers

    What is the condition of orthogonality for two circles x^2 + y^2 = a^2 and x^2 + y^2 = b^2?

    <p>a^2 + b^2 = (α - β)^2</p> Signup and view all the answers

    What is the equation of the circle formed by the intersection of two circles x^2 + y^2 = a^2 and x^2 + y^2 = b^2?

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

    What is the centre-distance relationship for a point P(x_1, y_1) and a circle x^2 + y^2 + 2gx + 2fy + c = 0?

    <p>|(x_1 + g)^2 + (y_1 + f)^2 - c| = R</p> Signup and view all the answers

    What is the condition for two circles to intersect at a right angle?

    <p>The tangents at the point of intersection are perpendicular</p> Signup and view all the answers

    If two circles intersect at a point, what can be said about the tangents at that point?

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

    What is the equation of the circle formed by the intersection of two circles x^2 + y^2 = a^2 and x^2 + y^2 = b^2, where a ≠ b?

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

    What is the relationship between the centre and the point of intersection of two circles, if the distance between their centres is equal to the sum of their radii?

    <p>The centre is equidistant from the point of intersection</p> Signup and view all the answers

    If two circles have two direct common tangents that are real and distinct, what can be said about the distance between their centres?

    <p>It is greater than the difference of their radii</p> Signup and view all the answers

    When the distance between the centres of two circles is equal to the sum of their radii, how many direct common tangents can be drawn?

    <p>Two</p> Signup and view all the answers

    What is the condition for two circles to intersect orthogonally?

    <p>The product of their radii is equal to half the distance between their centres</p> Signup and view all the answers

    What is the equation of the circle formed by the intersection of two circles?

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

    When the distance between the centres of two circles is less than the difference of their radii, what can be said about the tangents?

    <p>Two direct common tangents are real and coincident</p> Signup and view all the answers

    What is the centre-distance relationship when the distance between the centres of two circles is less than the sum of their radii?

    <p>|C1 C2| &lt; r1 + r2</p> Signup and view all the answers

    If two circles intersect at an angle of 90 degrees, what can be said about the centres of these circles?

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

    What is the condition of orthogonality for two circles x^2 + y^2 = a^2 and x^2 + y^2 = b^2?

    <p>a - b = 0</p> Signup and view all the answers

    What is the equation of the circle formed by the intersection of two circles x^2 + y^2 = a^2 and x^2 + y^2 = b^2?

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

    If two circles have three direct common tangents, what can be said about the distance between their centres?

    <p>It is equal to the difference of their radii.</p> Signup and view all the answers

    What is the centre-distance relationship for a point P(x_1, y_1) and a circle x^2 + y^2 + 2gx + 2fy + c = 0?

    <p>g(x_1) + f(y_1) = c</p> Signup and view all the answers

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