Grade 12 Maths · Analytical Geometry · Decide if a line cuts a circle

Decide if a line cuts a circle Grade 12

Substitute the line into the circle equation, then use the discriminant to say whether the line cuts the circle twice, touches it (a tangent) or misses it, or to find the value of k that makes it a tangent.

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Does Line Intersect Circle Grade 12Kevinmathscience

Finding the Points of intersection between the Circle and a LineShannon Ahern

A* Discriminant- Tangent to a Circle- A Level MathsExcel In Maths

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Question 4.3 · November 2022 · Paper 2 · National (DBE) · 4 marks

In the diagram,

M (3; - 5)

is the centre of the circle having PN as its diameter. KL is a tangent to the circle at

N (7; - 2)

. $Diagram for this question$

4.3 For which values of

k

will

y = - \frac{4}{3} x + k

be a secant to the circle?(4)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More questions: decide if a line cuts a circle

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3. Question 4.4 · September 2023 · Paper 2 · Provincial · 6 marks
In the diagram below, a circle, centered at M $(p; q)$ , touches the $x$ -axis at S and the line OA is a tangent to the circle at N $(6; - 8)$ . $Diagram for this question$
4.4 The line $y = - \frac{4}{3} x + t$ cuts the circle at two different points. Determine the values of $t$ .(6)
Show the memo (6 ticks)Memo
1. ✓
  Formula and substitutionLet C be the point on the circle directly opposite N (NC is a diameter).
  C: $\frac{x + 6}{2} = 10; \frac{y - 8}{2} = - 5$
  1 mark · Calculate the mid-point of a segment
2. ✓
  x-coordinate $x = 14$
  1 mark · Calculate the mid-point of a segment
3. ✓
  y-coordinate $y = - 2$
  $C (14; - 2)$
  1 mark · Calculate the mid-point of a segment
4. ✓
  SubstitutionEquation of the tangent at C:
  $y + 2 = - \frac{4}{3} (x - 14)$
  1 mark · Find the equation of a line
5. ✓
  Answer $y = - \frac{4}{3} x + \frac{50}{3}$
  1 mark · Find a tangent to a circle
6. ✓
  Value of t $∴ 0 < t < \frac{50}{3}$
  1 mark · Decide if a line cuts a circle
4. Practice · NavyBlue · 2 marks
The circle with centre $C$ has equation $(x - 1)^{2} + (y - 1)^{2} = 8$ . The line $y = - x + 6$ is drawn.
(b) What does your answer to (a) tell you about the line? Explain.(2)
Show the memo (2 ticks)Memo
1. ✓
  InterpretThere is only one (repeated) solution, $(3; 3)$ : the line meets the circle at exactly one point, so it is a tangent to the circle at $(3; 3)$ .
  2 marks · Decide if a line cuts a circle
5. Practice · NavyBlue · 2 marks
The line $y = x + 4$ is a tangent at $T$ to a circle with centre $M (3; - 1)$ .
(d) Confirm, by solving the two equations together, that $y = x + 4$ touches the circle at exactly one point.(2)
Show the memo (2 ticks)Memo
1. ✓
  Substitute $(x - 3)^{2} + (x + 5)^{2} = 32$ gives $2 x^{2} + 4 x + 2 = 0$ , that is $x^{2} + 2 x + 1 = 0$
  1 mark · Solve linear and non-linear simultaneous equations
2. ✓
  Discriminant $Δ = 2^{2} - 4 (1) (1) = 0$ : equal roots, so the line meets the circle at one point only ( $x = - 1$ )
  1 mark · Decide if a line cuts a circle
6. Practice · NavyBlue · 2 marks
Consider the circle $x^{2} + y^{2} = 16$ and the line $y = x + 6$ .
(b) Calculate the discriminant and interpret it.(2)
Show the memo (2 ticks)Memo
1. ✓
  Discriminant $Δ = 6^{2} - 4 (1) (10) = - 4$
  1 mark · Calculate the discriminant
2. ✓
  Interpret $Δ < 0$ : there are no real roots, so the line and the circle do not meet.
  1 mark · Decide if a line cuts a circle

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