Lesson: Tangents and Normals to Graphs of Implicit Functions

In this lesson, we will learn how to find the tangent and normal lines to a function implicitly.

Sample Question Videos

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  • 08:34

Worksheet: Tangents and Normals to Graphs of Implicit Functions • 25 Questions • 2 Videos

Q1:

Determine the points on the curve π‘₯ + 𝑦 + π‘₯ + 8 𝑦 = 0 2 2 at which the tangent is perpendicular to the line 7 𝑦 + 4 π‘₯ + 𝑐 = 0 .

Q2:

Find the equation of the tangent to the curve 9 π‘₯ βˆ’ 6 π‘₯ + 6 π‘₯ βˆ’ 𝑦 βˆ’ 𝑦 + 2 = 0 3 2 2 at the point ( 0 , 1 ) .

Q3:

Do the curves 9 𝑦 βˆ’ 8 𝑦 = 6 π‘₯ 4 and βˆ’ 5 π‘₯ βˆ’ 3 𝑦 = βˆ’ 4 π‘₯ 2 intersect orthogonally at the origin?

Q4:

The tangent at ( βˆ’ 2 , 2 ) to the curve π‘₯ + π‘₯ 𝑦 + 5 π‘₯ + 5 𝑦 = 0 3 2 makes a positive angle with the positive π‘₯ -axis. Find this angle.

Q5:

The tangent at ( 1 , 1 ) to the curve 5 π‘₯ 𝑦 + 2 𝑦 π‘₯ = 7 makes a positive angle with the positive π‘₯ -axis. Find this angle.

Q6:

Determine the slope of the tangent to π‘₯ βˆ’ 𝑦 + 9 π‘₯ + 2 𝑦 + 8 = 0 2 2 at its intersection with the π‘₯ -axis.

Q7:

Find the points of βˆ’ π‘₯ + 2 𝑦 = βˆ’ 4 2 2 where the angle between the tangent and the positive π‘₯ -axis has cosine 4 5 .

Q8:

Find the points on the curve 5 π‘₯ βˆ’ 8 π‘₯ 𝑦 + 4 𝑦 = 4 2 2 at which the tangent is parallel to the 𝑦 axis.

Q9:

Find the points that lie on the curve 2 π‘₯ βˆ’ π‘₯ 𝑦 + 2 𝑦 βˆ’ 4 8 = 0 2 2 at which the tangent is parallel to line 𝑦 = βˆ’ π‘₯ .

Q10:

Determine the points on a curve π‘₯ + 𝑦 = 4 5 2 2 at which the tangent to the curve is perpendicular to the straight line 𝑦 = 2 π‘₯ + 1 2 .

Q11:

Find the equation of the tangent to the curve π‘₯ + 𝑦 + 4 π‘₯ βˆ’ 5 𝑦 βˆ’ 5 = 0 2 2 at the point ( 1 , 0 ) .

Q12:

Find the equation of the tangent to the curve at the point .

Q13:

Find the equation of the tangent to the curve 𝑦 = 4 π‘₯ + 4 π‘₯ + 3 3 2 at π‘₯ = 2 .

Q14:

Find the equation of the tangent to the curve π‘₯ 6 𝑦 = βˆ’ 8 𝑦 6 π‘₯ s i n c o s at ο€» πœ‹ 4 , πœ‹ 2  .

Q15:

At the point ( 0 , βˆ’ 2 ) , determine the equation of the normal to the curve represented by the equation 6 π‘₯ + 2 π‘₯ + 2 π‘₯ βˆ’ 9 𝑦 βˆ’ 8 𝑦 + 2 0 = 0 3 2 2 .

Q16:

Find the equation of the normal to the curve π‘₯ 𝑦 βˆ’ 4 π‘₯ + 2 𝑦 βˆ’ 2 0 = 0 2 2 at the point ( 1 , 4 ) .

Q17:

Find the equation of the normal to the curve of the function π‘₯ 6 𝑦 = βˆ’ 6 𝑦 6 π‘₯ s i n c o s at ο€» πœ‹ 4 , πœ‹ 2  .

Q18:

A tangent to π‘₯ + 𝑦 = 7 2 2 2 forms an isosceles triangle when taken with the positive π‘₯ - and 𝑦 -axes. What is the equation of this tangent?

Q19:

At a point on the curve π‘₯ + 3 π‘₯ + 𝑦 + 5 𝑦 + 4 = 0 2 2 with π‘₯ < 0 , 𝑦 < 0 , the tangent makes an angle of 9 πœ‹ 4 with the positive π‘₯ -axis. Find the equation of the tangent at that point.

Q20:

At a point on the curve π‘₯ + 5 π‘₯ + 𝑦 βˆ’ 𝑦 βˆ’ 6 = 0 2 2 with π‘₯ < 0 , 𝑦 < 0 , the tangent makes an angle of 7 πœ‹ 4 with the positive π‘₯ -axis. Find the equation of the normal at that point.

Q21:

The point ( βˆ’ 5 , βˆ’ 2 ) lies on the curve π‘₯ + 𝑦 βˆ’ 3 π‘˜ π‘₯ + 7 = 0 2 2 . Find π‘˜ and the equation of the tangent to the curve at this point.

Q22:

Find the area of the triangle bounded by the π‘₯ -axis, the tangent, and the normal to the curve π‘₯ + 5 𝑦 = 1 5 2 2 at the point ( 9 , 2 ) to the nearest thousandth.

Q23:

Find the equations of the tangent lines of the curves and at the intersection points, and state whether the curves intersect orthogonally or not.

Q24:

The two curves ( π‘₯ βˆ’ π‘Ž ) + 𝑦 = 5 0   and ( π‘₯ + π‘Ž ) + 𝑦 = 5 0   intersect orthogonally. Find all the possible values of π‘Ž .

Q25:

Find the equation of the tangent to 9 𝑦 = βˆ’ 7 π‘₯ + 9 2 that has slope 7 1 8 .

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