Search any question & find its solution
Question:
Answered & Verified by Expert
The tangent drawn at any point $P$ to the parabola $y^2=4 a x$ meets the directrix at the point $K$, then the angle which $K P$ subtends at its focus is
Options:
Solution:
2086 Upvotes
Verified Answer
The correct answer is:
$90^{\circ}$
Here, $P\left(a t^2, 2 a t\right)$ and $S(a, 0)$.
If the tangent at $P, t y=x+a t^2$, meets the directrix
$x=-a$ at $k$, then $k=\left(-a, \frac{a t^2-a}{t}\right)$
$m_1=$ slope of $S P-\frac{2 a t}{a\left(t^2-1\right)}$
$m_2=$ slope of $S K=\frac{a\left(t^2-1\right)}{-2 a t}$
Clearly $m_1 m_2=-1, \quad \therefore \angle P S K=90^{\circ}$.
If the tangent at $P, t y=x+a t^2$, meets the directrix
$x=-a$ at $k$, then $k=\left(-a, \frac{a t^2-a}{t}\right)$
$m_1=$ slope of $S P-\frac{2 a t}{a\left(t^2-1\right)}$
$m_2=$ slope of $S K=\frac{a\left(t^2-1\right)}{-2 a t}$
Clearly $m_1 m_2=-1, \quad \therefore \angle P S K=90^{\circ}$.
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.