NIMCET 2023 — Mathematics PYQ

Correct Answer - Option 2: $y=mx-a\left(m+\frac{1}{m}\right)$

CONCEPT:
- If x = f(t), y = g(t), where t is a parameter, then $\frac{dy}{dx}=\frac{g^{\prime }(t)}{f^{\prime }(t)}$
- Tangent to a curve y = f(x) at any point is given by: m = dy/dx
- Equation of a tangent to a curve y = f(x) is given by: y - y_1 = m × (x - x_1)

CALCULATION:
Given: Equation of curve is x = a cos 2t, y = 2√2 a sin t

As we know that, tangent to a curve y = f(x) at any point is given by: m = dy/dx
Here, $m=\frac{dy}{dx}=-\frac{\sqrt{2}}{2\sin t}$

$\Rightarrow \sin t=-\frac{\sqrt{2}}{2m}$

As it is given that, x = a cos 2t, y = 2√2 a sin t

So, by using the value of sin t in x = a cos 2t, y = 2√2 a sin t we get

$\Rightarrow y=-\frac{2a}{m}$ and $x=a\left(1-\frac{1}{m^2}\right)$

As we know that, equation of a tangent to a curve y = f(x) is given by: y - y_1 = m × (x - x_1)

Here, any point on the curve is given by: $\left(a\left(1-\frac{1}{m^2}\right),-\frac{2a}{m}\right)$

So, the equation of the required tangent is: y + \frac{2a}{m} = m × \left[ x - a \left( 1 - \frac{1}{m^2} \right) \right]

On further simplifying the above equation we get,

$\Rightarrow y=mx-a\left(m+\frac{1}{m}\right)$

Hence, option B is the correct answer.