Cards In This Deck

• $\text{What is the first step to determine if a specific point }(p,q)\text{ lies inside, outside, or on a circle with equation }(x-a{)}^2+(y-b{)}^2=r^2\text{?}$
  $\text{Calculate the distance squared from the centre to the point: }(p-a{)}^2+(q-b{)}^2\text{ and compare it to }{r}^2\text{.}$
• $\text{When solving for the intersection of a line }y=mx+c\text{ and a circle, you arrive at a quadratic equation. What does it mean if the discriminant (}{b}^2-4ac\text{) of this quadratic is zero?}$
  $\text{The line is a tangent to the circle.}$
• $$\begin{gathered} \text{True or False?} \\ \text{To find the midpoint of a line segment with endpoints }(x_1,y_1)\text{ and }(x_2,y_2)\text{, you calculate }(\frac{x_1-x_2}{2},\frac{y_1-y_2}{2})\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Formula: }(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\text{.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{If the equation of a circle is rearranged to }(x-3{)}^2+(y+4{)}^2=25\text{, the radius is 25.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: The radius is }5\text{.} \end{gathered}$$
• $\text{What is the standard procedure to find the equation of a tangent to a circle at a specific point }P(x_1,y_1)\text{?}$
  $$\begin{gathered} \text{1. Find the gradient of the radius (}{m}_r\text{) from the centre to }P\text{.} \\ \text{2. Calculate the gradient of the tangent (}{m}_t\text{) using }{m}_t=-\frac{1}{m_r}\text{.} \\ \text{3. Use }y-y_1=m_t(x-x_1)\text{ to find the equation.} \end{gathered}$$
• $\text{What is the relationship between the perpendicular bisector of a chord and the centre of the circle?}$
  $\text{The perpendicular bisector of any chord always passes through the centre of the circle.}$
• $\text{If three points }A\text{, }B\text{, and }C\text{ lie on a circle, what condition confirms that line segment }AC\text{ is the diameter?}$
  $\text{If the angle }\angle ABC=90^{\circ }\text{.}$
• $\text{Given three points on a circle’s circumference, which geometric property provides the most efficient method to find the circle’s centre?}$
  $$\begin{gathered} \text{The perpendicular bisector of any chord passes through the centre.} \\ \text{Find the equations of the perpendicular bisectors of two different chords (e.g., AB and BC) and find their point of intersection.} \end{gathered}$$
• $\text{You are given the equation of a circle }{x}^2+y^2+2fx+2gy+c=0\text{. How do you quickly identify the coordinates of the centre without full completion of the square?}$
  $\text{The centre is }(-f,-g)\text{.}$
• $$\begin{gathered} \text{True or False?} \\ \text{To find the centre and radius of the circle given by }2x^2+2y^2-8x+12y-6=0\text{, you should immediately start completing the square on }2x^2-8x\text{ and }2y^2+12y\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Step: You must divide the entire equation by the coefficient of }{x}^2\text{ (which is 2) before completing the square.} \end{gathered}$$