Question 1 Easy
The radius to (5, 12) has which gradient?
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Answer: 12/5
- y divided by x.
- 12/5.
- Method (1): y divided by x.
- Accuracy (1): Answer 12/5
Equation of a tangent practice questions with answers
A second set of equation of a tangent questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.
More questions
Question 2 Easy
Does (6, 8) lie on x^2 + y^2 = 100? Answer yes or no.
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Answer: yes
- 36 + 64 = 100.
- The point is on the circle.
- Method (1): 36 + 64 = 100.
- Accuracy (1): Answer yes
Question 3 Medium
At (-3, 4) on x^2 + y^2 = 25, what is the tangent gradient?
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Answer: 3/4
- Radius gradient 4/(-3) = -4/3.
- Negative reciprocal is 3/4.
- Method (1): Radius gradient 4/(-3) = -4/3.
- Accuracy (1): Answer 3/4
Question 4 Medium
Write the tangent to x^2 + y^2 = 25 at (0, 5).
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Answer: y=5
- The radius is vertical, so the tangent is horizontal.
- y = 5.
- Method (1): The radius is vertical, so the tangent is horizontal.
- Accuracy (1): Answer y=5
Question 5 Hard
A radius has gradient 2. What is the tangent gradient?
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Answer: -1/2
- Negative reciprocal of 2.
- -1/2.
- Method (1): Negative reciprocal of 2.
- Accuracy (1): Answer -1/2
Question 6 Hard
Does (5, 12) lie on x^2 + y^2 = 169? Answer yes or no.
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Answer: yes
- 25 + 144 = 169.
- It lies on the circle, so a tangent there is valid.
- Method (1): 25 + 144 = 169.
- Accuracy (1): Answer yes
Exam-style question
A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.
Exam-style question 14 marks
Find the equation of the tangent to the circle x^2 + y^2 = 25 at the point (3, 4).
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Answer: y=-3/4x+25/4
- Radius gradient 4/3.
- Tangent gradient -3/4.
- y - 4 = -3/4 (x - 3).
- y = -3/4 x + 25/4.
- Method (1): Radius gradient
- Method (1): Perpendicular gradient
- Method (1): Line through the point
- Accuracy (1): Equation
Quick checks
Short one-step questions. Pick an option before you open the answer.
Check 1
Gradient of the radius from the origin to (x, y)?
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y/x. It is the rise over the run from the origin. y/x. It is the rise over the run from the origin.
Check 2
How do you get the tangent gradient from the radius gradient?
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Take the negative reciprocal. The product of the gradients is -1. Take the negative reciprocal. The product of the gradients is -1.
Check 3
What is the negative reciprocal of 4/3?
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-3/4. Flip the fraction and change the sign. -3/4. Flip the fraction and change the sign.
Check 4
Which point do you substitute into y - y1 = m(x - x1)?
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The point where the tangent touches the circle. The point where the tangent touches the circle.
Check 5
Why is the tangent at (0, 5) the line y = 5?
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The radius is vertical, so the tangent is horizontal through that point. The radius is vertical, so the tangent is horizontal through that point.
Check 6
Why check x^2 + y^2 = r^2 first?
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The formula is for a point on the circle. A point off the circle is not a point of contact. The formula is for a point on the circle. A point off the circle is not a point of contact.
Check 7
The tangent at (3, 4) to x^2 + y^2 = 25 is which equation?
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y = -3/4 x + 25/4. y = -3/4 x + 25/4.
Check 8
Why is the intercept 25/4 and not 4?
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Rearranging y - 4 = -3/4 (x - 3) produces 25/4, which is r^2 divided by the y-coordinate. Rearranging y - 4 = -3/4 (x - 3) produces 25/4, which is r^2 divided by the y-coordinate.
Check 9
Gradient of the radius to (5, 12)?
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12/5. This is y divided by x. 12/5. This is y divided by x.
Check 10
Tangent gradient at (5, 12)?
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-5/12. It is the negative reciprocal of 12/5. -5/12. It is the negative reciprocal of 12/5.
Check 11
What angle does the radius make with the tangent?
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90 degrees. 90 degrees.
Check 12
Negative reciprocal of -4/3?
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3/4. Flip the fraction and change the sign. 3/4. Flip the fraction and change the sign.
Check 13
Can two different points on the same circle share a tangent gradient?
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Yes. (3, 4) and (-3, -4) on x^2 + y^2 = 25 both give tangent gradient -3/4, but the intercepts are 25/4 and -25/4. Yes. (3, 4) and (-3, -4) on x^2 + y^2 = 25 both give tangent gradient -3/4, but the intercepts are 25/4 and -25/4.
Check 14
What is y/x at (8, 6)?
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6/8, which is 3/4. 6/8, which is 3/4.
Check 15
Write the first line of working for a tangent, before any algebra.
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Radius gradient, then negative reciprocal. Radius gradient, then negative reciprocal.
Keep going with Equation of a tangent
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