Question

if the tangents drawn from the point (0,2) to the parabola y^2 = 4ax are inclined at angle 3?/4 , then the value of a is.
Please don't do it by T^2=SS1

Answer 1

Tangents are drawn from the parabola$y^{2} =4 a x$  from the point (0,2).

Equation of line passing through (0,2) and having slope m

→ y -2 = m x

→ y = m x + 2

putting the value of y in equation of parabola $y^{2} =4 a x$

→ (m x + 2)² = 4 a x

→ m²x² + 4 + 4 m x=4 a x

→m²x² +4 m x-4 a x+4=0

→m²x² +4 x m- 4 a x+4=0

As this is a quadratic equation in m.So solution is given by

$m=\frac{-4 x \pm\sqrt {16 x^{2}-4x^{2}(4-4ax)}}{2x^{2}}$

$m=\frac{-4 x\pm4x \sqrt{xa}}{2 x^{2}}=\frac{-2\pm2\sqrt{xa}}{x}$

So, $m_{1}=\frac{-2 + 2\sqrt{ax}}{x} and m_{2}=\frac{-2 - 2\sqrt{ax}}{x}$

For roots to exist a>0,and x>0 or a<0 and x<0.

Let angle between two tangents be α.

$tan\alpha =\pm \frac{m_{1}-m_{2}}{1+m_{1}m_{2}}$

As given , α=3π/4

tan 3π/4=$\frac{4\sqrt{ax}}{5-4ax}$

$-1=\frac{4\sqrt{ax}}{5-4ax}$

Squaring both sides,we get

→16 ax =25+ 16a²x²-40ax

→ 16 a²x²-56 a x+25=0.....(1)

As this is quadratic in x .

$x=\frac{56a\pm\sqrt{3136a^{2}-1600a^{2}}}{32a^{2}}$

x= $\frac{56\pm\sqrt{1536}}{32a}$

which gives x as 95.2/32a or 16.8/32a  [approx]

As y²=4 a x, replacing equation 1 in terms of y

$y^{4}-14 y+25=0$

Solving by discriminant method

$y^{2}=\frac{14\pm\sqrt{96}}{2}$

which gives two values of y²=11.9 or 5.2(approx)

Substituting the values of x and y in the equation

y²= 4a x, we get L.H.S=R.H.S →which is an identity.

So there are infinite values of a i.e any real umber for which angle between two tangents is 3π/4 because x is dependent on a.

Answer 2

Thank you for asking this question. Here is your answer:

ty = x + at² at (at², 2at)

we know that it passes through (0,2)

2t = 0 + at²

at² - 2t = 0

t (at -2) = 0

t = 0 or 2/a

the point will be (0,0) (4/a,4)

tangents of x are equal to 0

and the other tangent has a slope

= 4-2/4/a-0 it will pass through (4/a,4) and (0,2)

= a/2

and the angle between them is 3π/4

and the acute angle which is formed will be π/4

so a/2 = tan (π/4) or tan (3π/4)

a/2 = + - 1

a = + - 2

If there is any confusion please leave a comment below.