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Sagot :
The equations of the tangents that pass through (2,−3) are y=−x−1 and y= 11x−25.
What is an equation?
In arithmetic,an equation may be a formula that expresses the equality of 2 expressions, by connecting them with the sign =.
Main BODY:
The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point. So for our curve (the parabola) we have
y=x²+x
Differentiating wrt x we get:
dy / dx = 2x +1
Let P(α,β) be any generic point on the curve. Then the gradient of the tangent at P is given by:
m=2α+1 (using the derivative)
And as P lies on the curve we also have:
β=α²+α (using the curve equation)
And so the tangent at P passes through (α,α²+α) and has gradient 2α+1, so using the point/slope form y−y₁=m(x−x₁) the equation of the tangent at P is;
y−(α²+α)=(2α+1)(x−α)
if this tangent also passes through
(2,−3) then;
⇒-3−(α²+α)=(2α+1)(2−α)
⇒-3 −α²- α = 3α−2α²+2
⇒α²−4α−5=0
∴α=−1,5
If α=−1⇒β=0
, and the tangent equation becomes:
y−0=(−1)(x+1)
∴ y=−x−1
If α=5⇒β=30
, and the tangent equation becomes:
y−30=(11)(x−5
∴y−30=11x−55
∴y=11x−25
Hence the equations of the tangents that pass through (2,−3) are y=−x−1 and y= 11x−25.
To know more about equation , visit:
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