Parabola Question - None

Question

The equation of a tangent to the parabola, x² = 8y, which makes an angle $\theta$ with the positive direction of X-axis, is

Accepted Answer

Given parabola is x² = 8y ...(i)
Now, slope of tangent at any point (x, y) on the parabola (i) is
$\frac{dy}{dx}=\frac{x}{4}=tan heta$
[$\because$ tangent is making an angle $ heta$ with the positive direction of X-axis]
So, x = t tan$ heta$
$\Rightarrow$ 8y = (4 tan$ heta$)²[on putting x = 4 tan$ heta$ in Eq. (i)]
$\Rightarrow$ y = 2 tan² $ heta$
Now, equation of required tangent is y - 2 tan² $ heta$ = tan $ heta$ (x - 4 tan$ heta$)
$\Rightarrow$ y = x tan$ heta$ - 2 tan² $ heta$ $\Rightarrow$ x = y cot $ heta$ + 2 tan $ heta$

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