If the tangent to y2=4 a x at the point (a t2, 2 a t) where |t|1 is a normal to x2-y2=a2 at the point (a sec θ, a tan θ), then

Question

If the tangent to y2=4 a x at the point (a t2, 2 a t) where |t|>1 is a normal to x2-y2=a2 at the point (a sec θ, a tan θ), then (A) t = - textcosec θ (B) t = - sec θ (C) t = 2 tan θ (D) t = 2 cot θ

Tardigrade Admin · Accepted Answer

Equation of tangent to y2=4 a x at (a t2, 2 a t) will be x-y t=-a t2 ...(i) Also, equation of normal to x2-y2=a2 at (a sec θ, a tan θ) wiil be (x/a sec θ)+(y/a tan θ)=2 ⇒ x+y textcosec θ=2 a sec θ ...(ii) Since, Eqs. (i) and (ii) are identical. ∴ t=- textcosec θ or t=2 tan θ

Q. If the tangent to $y^{2} = 4 a x$ at the point $(a t^{2}, 2 a t)$ where $∣ t ∣ > 1$ is a normal to $x^{2} - y^{2} = a^{2}$ at the point $(a sec θ, a tan θ)$ , then

$t = - cosec θ$

$t = - sec θ$

$t = 2 tan θ$

$t = 2 cot θ$

Q. If the tangent to y2=4ax at the point (at2,2at) where ∣t∣>1 is a normal to x2−y2=a2 at the point (asecθ,atanθ), then

t=−cosecθ

t=−secθ

t=2tanθ

t=2cotθ

Equation of tangent to y2=4ax at (at2,2at) will be x−yt=−at2...(i) Also, equation of normal to x2−y2=a2 at (asecθ,atanθ) wiil be asecθx​+atanθy​=2 ⇒x+ycosecθ=2asecθ...(ii) Since, Eqs. (i) and (ii) are identical. ∴t=−cosecθ or t=2tanθ

Q. If the tangent to $y^{2} = 4 a x$ at the point $(a t^{2}, 2 a t)$ where $∣ t ∣ > 1$ is a normal to $x^{2} - y^{2} = a^{2}$ at the point $(a sec θ, a tan θ)$ , then

$t = - cosec θ$

$t = - sec θ$

$t = 2 tan θ$

$t = 2 cot θ$