Question

Find the number of distinct real tangent that can be drawn from (0,-2) to parabola y^2=4x . Also find slope of tangents

Answer 1

Why can't u understand above
It is the method and it is lengthy try to understand it and no other way to find in 2 or 4 steps it is quite difficult and time consuming problem not like other because it is exemplary and brainlist one


Answers
y² = 4x
x = 1/4 y²
m = (y + 2) / (x + 0):

m = (y + 2) / x


Differentiating y² = 4x,

2y(dy/dx) = 4
dy/dx = 4 / 2y
dy/dx = 2/y


m = dy/dx, so

(y + 2) / x = 2 / y
y + 2 = 2x / y
y(y + 2) = 2x
y² + 2y = 2x
x = 1/2 (y² + 2y)
x = 1/2 y² + y


x = x, so

1/4 y² = 1/2 y² + y
4(1/4 y²) = 4(1/2 y²) + 4(y)
y² = 2y² + 4y
y² - 2y² - 4y = 0
- y² - 4y = 0
y² + 4y = 0
y(y + 4) = 0

If the product of two factors equals zero,
then one or both factors equal zero.

If y = 0,

x = 1/4(0)²
x = 0

One Point of Tangency (0, 0)

Since the x-value of the point is zero and the x-value of the point of tangency is zero, then

Equation of One Tangent:

x = 0
¯¯¯¯¯

If y + 4 = 0,
y = - 4

and

x = 1/4 (- 4)²
x = 1/4 (16)
x = 4

Other point of Tangency (4, - 4)


x₁ = 0
y₁ = - 2

x₂ = 4
y₂ = - 4

m = (y₂ - y₁) / (x₂ - x₁):

m = [- 4 - (- 2)] / (4 - 0)
m = (- 4 + 2) / 4
m = - 2 / 4
m = - 1/2

b = y₁ - m(x₁):

b = - 2 - [- 1/2(0)]
b = - 2

Equation of Other Tangent LIne:

y = - 1/2 x - 2