The normal at 3 points P-Q-R on y2-4ax meet at a point N- If S is focus of parabola then the value ofSP-SQ-SR-SAMN is where A is vertex of parabola M is the foot of perpendicular from N on to tangent at vertex

Let N=(h,k) Normal is y= tx+2at+at^3. It is passing through (h,k) ⇒ k= th+2at+at^3 ⇒ at^3+(2a h)t k=0 It has 3 roots ⇒ t_1,t_2,t_3 ⇒ t_1+t_2+t_3=0...(i) t_1t_2+ ...

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Standard XI

Mathematics

Equation of Normal at Given Point

The normal at...

Question

The normal at $3$ points $P, Q, R$ on $y^{2} = 4 a x$ meet at a point $N .$ If $S$ is focus of parabola then the value of $\frac{S P + S Q + S R + S A}{M N}$ is
(where $A$ is vertex of parabola $M$ is the foot of perpendicular from $N$ on to tangent at vertex)

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Solution

Let $N = (h, k)$
Normal is $y = - t x + 2 a t + a t^{3} .$
It is passing through $(h, k)$
$\Rightarrow k = - t h + 2 a t + a t^{3}$
$\Rightarrow a t^{3} + (2 a - h) t - k = 0$
It has $3$ roots $\Rightarrow t_{1}, t_{2}, t_{3}$
$\Rightarrow t_{1} + t_{2} + t_{3} = 0... (i)$
$t_{1} t_{2} + t_{2} t_{3} + t_{3} t_{1} = \frac{2 a - h}{a} . . . (i i)$
$t_{1} t_{2} t_{3} = \frac{k}{a} . . . (i i i)$
Now $P = (a t_{1}^{2}, 2 a t_{1}), Q (a t_{2}^{2}, 2 a t_{2}), R (a t_{3}^{2}, 2 a t_{3})$
$S = (a, 0), A (0, 0), M = (0, k), N = (h, k)$
Now,
$\frac{S P + S Q + S R + S A}{M N} = \frac{a + a t_{1}^{2} + a + a t_{2}^{2} + a + a t_{3}^{2} + a}{h}$
$= \frac{4 a + a (t_{1}^{2} + t_{2}^{2} + t_{3}^{2})}{h}$
$= \frac{4 a + a {(t_{1} + t_{2} + t_{3})^{2} - 2 (t_{1} t_{2} + t_{2} t_{3} + t_{3} t_{1})}}{h}$
Putting values from $(i)$ and $(i i)$ , we get
$\frac{S P + S Q + S R + S A}{M N} = 2$

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Equation of Normal at Given Point

Standard XI Mathematics

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The normal at 3 points P,Q,R on y2=4ax meet at a point N. If S is focus of parabola then the value ofSP+SQ+SR+SAMN is (where A is vertex of parabola M is the foot of perpendicular from N on to tangent at vertex)

The normal at $3$ points $P, Q, R$ on $y^{2} = 4 a x$ meet at a point $N .$ If $S$ is focus of parabola then the value of $\frac{S P + S Q + S R + S A}{M N}$ is
(where $A$ is vertex of parabola $M$ is the foot of perpendicular from $N$ on to tangent at vertex)