find the equation of tangent to the parabola y^2 = 9x at ( 1 ,- 3 ) . please solve this .
Answers
EXPLANATION.
Equation of tangent to the parabola : y² = 9x.
At the points = (1,-3).
As we know that,
General equation of parabola : y² = 4ax.
Compare both the equation, we get.
⇒ 4a = 9.
⇒ a = 9/4.
As we know that,
Formula of :
The equation of the tangent of the parabola y² = 4ax is y = mx + a/m.
Put the values of (x, y) = (1,-3),
⇒ (-3) = m(1) + 9/4m.
⇒ - 3 = m + 9/4m.
⇒ - 12m = 4m² + 9.
⇒ 4m² + 12m + 9 = 0.
Factorizes the equation into middle term splits, we get.
⇒ 4m² + 6m + 6m + 9 = 0.
⇒ 2m(2m + 3) + 3(2m + 3) = 0.
⇒ (2m + 3)(2m + 3) = 0.
⇒ (2m + 3)² = 0.
⇒ 2m + 3 = 0.
⇒ m = -3/2.
As we know that,
Formula of :
Equation of tangent.
⇒ (y - y₁) = m(x - x₁).
⇒ (y - (-3)) = -3/2(x - 1).
⇒ (y + 3) = -3/2(x - 1).
⇒ 2(y + 3) = -3(x - 1).
⇒ 2y + 6 = - 3x + 3.
⇒ 2y + 3x + 6 - 3 = 0.
⇒ 2y + 3x + 3 = 0.
MORE INFORMATION.
Conditions of tangency.
(1) = The line y = mx + c touches a parabola y² = 4ax then c = a/m.
(2) = The line y = mx + c touches a parabola x² = 4ay then c = -am².