if two tangents drawn from the point P to the parabola y2=12x be such that the slope of the tangent is double the other, then P lies on the curve
Answers
Answer:
2y2=27x
Solution....
Given a parabola ,y2=12x
There is a point P outside the parabola with coordinates(h,k).Now the equation of a tangent to the parabola y2=4ax is given by
y=mx+ma;m is the slope of tangent
Here a=3
y=mx+m3
is the equation of any tangent to the parabola y2=12x.This tangent passes through point (h,k) hence
⇒m2h−km+3=0−(i)
So we can see that there are two possible values of ′m′ and hence we can have two tangents drawn to the parabola y2=12x from the point (h,k).
Also the slope of one is twice that of the other.
So say m1 & m2 are the roots of the quadratic equation
Then we have
m1=2m2−(ii)
m1+m2=hk-(iii)
m1.m2=h3−(iv)
2m22=h3−(v)
3m2=hk⇒m2=3hk−(vi)
Using (vi) we can reduce (v) as 9h22k2=h3
⇒2k2=27h
Replace k with y & h with x we have
2y2=27x
EXPLANATION.
Two tangents drawn at a point ''P''.
To the parabola : y² = 12x.
Slope of tangent is double the other.
As we know that,
Concept of equation of tangent.
Equation of line : y = mx + c.
Equation of parabola : y² = 4ax.
Take two points on the parabola : (at², 2at).
Put the values of (at², 2at) in equation of line, we get.
⇒ 2at = mat² + c.
⇒ mat² - 2at + c = 0.
⇒ D = 0 Or b² - 4ac = 0.
⇒ (- 2a)² - 4(ma)(c) = 0.
⇒ 4a² - 4mac = 0.
⇒ a² - mac = 0.
⇒ a(a - mc) = 0.
⇒ c = a/m.
Put the value of c = a/m in equation of line, we get.
y = mx + a/m : Equation of tangents.
If tangents passes through point (h, k) then,
⇒ k = mh + a/m.
⇒ hm² - km + a = 0.
Using this concept in the equation, we get.
Equation of parabola : y² = 12x.
⇒ 4a = 12.
⇒ a = 3.
Put the value of a = 3 in equation of tangent, we get.
⇒ y = mx + 3/m.
⇒ k = mh + 3/m.
⇒ hm² - km + 3 = 0.
If m₁ and m₂ are the roots of the equation then,
Sum of the zeroes of the quadratic polynomial.
⇒ m₁ + m₂ = k/h. - - - - - (1).
Products of the zeroes of the quadratic polynomial.
⇒ m₁ m₂ = 3/h. - - - - - (2).
Slope of the tangent is double the other.
⇒ m₁ = 2m₂
Put the values of m₁ = 2m₂ in equation (), we get.
⇒ 2m₂ + m₂ = k/h.
⇒ 3m₂ = k/h.
⇒ m₂ = k/3h. - - - - - (3).
Put the values of m₁ = 2m₂ in equation (2), we get.
⇒ (2m₂) x (m₂) = 3/h.
⇒ (2m₂)² = 3/h. - - - - - (4).
Put the values of equation (3) in equation (4), we get.
⇒ 2(k/3h)² = 3/h.
⇒ 2(k²/9h²) = 3/h.
⇒ 2k²/9h² = 3/h.
⇒ 2k²h = 27h².
⇒ 2k² = 27h.
⇒ 2y² = 27x.
P lies on the curves : 2y² = 27x.