Circles: points, tangents and lines questions with answers

Questions

1. Question 1Inside, on or outside?

   Determine whether the point $(1, 5)$ lies inside, on or outside the circle $(x + 1)^2 + (y - 2)^2 = 10$.

   Show the worked solution

   1. Step 1: Work out (x − a)² + (y − b)² at the point$\mathstrut (1 - (-1))^2 + (5 - 2)^2$$\text{Value} = 13$
   2. Step 2: Compare it with r²$13 > 10$, so it is outside.
2. Question 2Inside, on or outside?

   Determine whether the point $(4, -1)$ lies inside, on or outside the circle $(x - 3)^2 + (y + 2)^2 = 25$.

   Show the worked solution

   1. Step 1: Work out (x − a)² + (y − b)² at the point$\mathstrut (4 - 3)^2 + (-1 - (-2))^2$$\text{Value} = 2$
   2. Step 2: Compare it with r²$2 < 25$, so it is inside.
3. Question 3Tangent at a point

   The point $P(6, 14)$ lies on the circle $(x - 1)^2 + (y - 2)^2 = 169$. Find the equation of the tangent to the circle at $P$.

   Show the worked solution

   1. Step 1: The radius goes from the centre to P; the tangent is perpendicular to it. Put the coordinates into the gradient formula$m_{\text{radius}} = \dfrac{y_2 - y_1}{x_2 - x_1}$$m_{\text{radius}} = \dfrac{14 - 2}{6 - 1}$$= \dfrac{12}{5}$$m_{\text{tangent}} = -\dfrac{5}{12}$
   2. Step 2: y − y₁ = m(x − x₁) through P, then make y the subject$y = -\dfrac{5}{12}x + \dfrac{33}{2}$
4. Question 4Tangent at a point

   The point $P(1, 2)$ lies on the circle $x^2 + y^2 = 5$. Find the equation of the tangent to the circle at $P$.

   Show the worked solution

   1. Step 1: The radius goes from the centre to P; the tangent is perpendicular to it. Put the coordinates into the gradient formula$m_{\text{radius}} = \dfrac{y_2 - y_1}{x_2 - x_1}$$m_{\text{radius}} = \dfrac{2 - 0}{1 - 0}$$= 2$$m_{\text{tangent}} = -\dfrac{1}{2}$
   2. Step 2: y − y₁ = m(x − x₁) through P, then make y the subject$y = -\dfrac{1}{2}x + \dfrac{5}{2}$
5. Question 5The number of points where a line meets it

   Find the number of points at which the line $y = -x$ meets the circle $x^2 + y^2 = 10$.

   Show the worked solution

   1. Step 1: Substitute the line into the circle, expand, and make one side zero$x^2 + (-x)^2 = 10$$x^2 + x^2 = 10$$2x^2 - 10 = 0$
   2. Step 2: The discriminant says how many points$b^2 - 4ac = (0)^2 - 4(2)(-10)$$= 80$It is positive, so the line meets the circle at two points.
6. Question 6The number of points where a line meets it

   Find the number of points at which the line $y = 2x + 10$ meets the circle $x^2 + y^2 = 25$.

   Show the worked solution

   1. Step 1: Substitute the line into the circle, expand, and make one side zero$x^2 + (2x + 10)^2 = 25$$x^2 + 4x^2 + 40x + 100 = 25$$5x^2 + 40x + 75 = 0$
   2. Step 2: The discriminant says how many points$b^2 - 4ac = (40)^2 - 4(5)(75)$$= 100$It is positive, so the line meets the circle at two points.