Question

Find an equation of the tangent line to the curve y^2-4y-8x-20=0 at its point of intersection with the line 2x+y+4=o

Answer 1

The curve is y ^2 - 4y - 8x - 20 = 0 and the line is 2x + y + 4 = 0.

Substitute the value of y = - 2x - 4 (2x + y + 4 = 0 → y = - 2x - 4) in y ^2 - 4y - 8x - 20 = 0.

(- 2x - 4)^2 - 4(- 2x - 4) - 8x - 20 = 0

4x ^2 + 16x + 16 + 8x + 16 - 8x - 20 = 0

4x ^2 + 16x + 12 = 0

x ^2 + 4x + 3 = 0

Factor by grouping.

x ^2 + 3x + x + 3 = 0

x(x + 3) + 1(x + 3) = 0

Factor : (x + 3)(x + 1) = 0

Apply zero product property.

x + 3 = 0 and x + 1 = 0

x = - 3 and x = - 1.

Substitute the values  of x = - 3 and x = - 1 in y = - 2x - 4.

y = - 2(- 3) - 4 = 2 and y = - 2(- 1) - 4 = - 2.

Therefore, the two points of intersection of the line and curve are (- 3, 2) and (- 1, - 2).

The curve is  y ^2 - 4y - 8x - 20 = 0 and the points are (- 3, 2) and (- 1, - 2).

We can use implicit differentiation to find y '.

The implicit (or total) derivation of y ^2 - 4y - 8x - 20 = 0 is,

2yy ' - 4y ' - 8 = 0

2y '(y - 2) = 8

y ' = 4 / (y - 2)

Case 1:

Substitute the values of (x, y ) = (- 3, 2) in the above equation.

y ' = 4 / (2 - 2)  = 4 / 0.

This is the slope (m ) of the tangent line to the implicit curve at (- 3, 2).

To find the tangent line equation, substitute the values of m = 4 / 0 and (x, y ) = (- 3, 2) in the slope intercept form of an equation.

y = mx + b

2 = (4 / 0)(- 3) + b

2 - b = - 12 / 0

- 12 = 0.

The above statement is false, so we can ' t find the tangent line through the point (- 3, 2).

Case 2:

Substitute the values of (x, y ) = (- 1, - 2) in y ' = 4 / (y - 2).

y ' = 4 / (- 2 - 2)

y ' = 4 / - 4 = - 1.

This is the slope (m ) of the tangent line to the implicit curve at (- 1, - 2).

To find the tangent line equation, substitute the values of m = - 1 and (x, y ) = (- 1, - 2) in the slope intercept form of an equation.

y = mx + b

- 2 = (- 1)(- 1) + b

b = - 2 - 1 = - 3.

Substitute m = - 1 and b = - 3 in y = mx + b.

y = (- 1)x + (- 3)

y = - x  - 3.

The tangent line equation is y = - x  - 3.