Unit 4
7 marks · HardA circle has equation . The point P(3, 4) lies on the circle.
Find the equation of the tangent to the circle at point P in the form .
If this tangent is written as , write down the value of m and interpret its meaning in the context of the circle.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) 3x + 4y = 25 · must show Equation of tangent in standard form
(b) m = -\frac{3}{4}; It is the negative reciprocal of the radius' gradient at point P, indicating perpendicularity. · must show Correct value and context interpretation
Also accept: Accept equations of tangent in any correct rearranged form
Common mistakes: Taking the radius as the tangent's gradient, not its negative reciprocal
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Finds gradient of radius OP as 4/3 (change in y over change in x from origin to P).
M2 (a) · 1 mark · Recognises tangent gradient is the negative reciprocal (-3/4).
M3 (a) · 1 mark · Uses point–gradient form to write tangent equation.
A1 (a) · 1 mark · Correctly simplifies to 3x + 4y = 25 or equivalent.
B1 (b) · 1 mark · Writes down m = -3/4 for the tangent's gradient.
B2 (b) · 1 mark · Explains that m is the negative reciprocal of the radius gradient (so tangent is perpendicular to the radius at P).
B3 (b) · 1 mark · Links to the context of the circle, i.e. the tangent only touches the circle at one point and is perpendicular to the radius at P.
Worked solution
a) Gradient of OP (radius) from centre (0,0) to P(3,4) is (4-0)/(3-0) = 4/3.
Gradient of tangent = -3/4 (negative reciprocal).
Multiply both sides by 4: 4y - 16 = -3(x - 3)
4y - 16 = -3x + 9
3x + 4y = 25
b) m = -3/4. This is the gradient of the tangent at P; it is perpendicular to the gradient of the radius at this point, matching the property that the tangent to a circle at any point is perpendicular to the radius at that point.