Let the point P of the focal chord PQ of the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The equation of the parabola is $y^2=16x$. Comparing this with $y^2=4ax$,we get $a=4$. The focus $S$ is $(a, 0) = (4, 0)$.Let the coordinates of p

Vedclass · Accepted Answer

$17$

Vedclass · Answer

(A) The equation of the parabola is $y^2=16x$. Comparing this with $y^2=4ax$,we get $a=4$. The focus $S$ is $(a, 0) = (4, 0)$.Let the coordinates of point $P$ be $(at_1^2, 2at_1)$. Given $P = (1, -4)$,we have $2at_1 = -4$ $\Rightarrow 2(4)t_1 = -4$ $\Rightarrow t_1 = -\frac{1}{2}$.Since $PQ$ is a focal chord,the product of the parameters of its endpoints is $t_1 t_2 = -1$. Thus,$t_2 = -\frac{1}{t_1} = -\frac{1}{-1/2} = 2$.The coordinates of point $Q$ are $(at_2^2, 2at_2) = (4(2)^2, 2(4)(2)) = (16, 16)$.Let the focus $S(4, 0)$ divide the chord $PQ$ in the ratio $m:n$. Using the section formula for the $x$-coordinate:$4 = \frac{m(x_Q) + n(x_P)}{m+n} = \frac{m(16) + n(1)}{m+n}$$4(m+n) = 16m + n$$4m + 4n = 16m + n$$3n = 12m \Rightarrow \frac{m}{n} = \frac{3}{12} = \frac{1}{4}$.Since $\operatorname{gcd}(m, n) = \operatorname{gcd}(1, 4) = 1$,we have $m=1$ and $n=4$.Therefore,$m^2+n^2 = 1^2+4^2 = 1+16 = 17$.

Let the point $P$ of the focal chord $PQ$ of the parabola $y^2=16x$ be $(1, -4)$. If the focus of the parabola divides the chord $PQ$ in the ratio $m:n$,where $\operatorname{gcd}(m, n)=1$,then $m^2+n^2$ is equal to:

Similar Questions

If $P$ is a point on the parabola $y^{2}=4ax$ with focus $F$. Let $Q$ denote the foot of the perpendicular from $P$ onto the directrix. Then, $\frac{\tan \angle PQF}{\tan \angle PFQ}$ is

The focal distance of a point on the parabola $y^{2}=16x$ whose ordinate is twice the abscissa,is

The locus of a point such that two tangents drawn from it to the parabola $y^2 = 4ax$ are such that the slope of one is double the other is:

How many parabolas can be drawn if the two endpoints of the latus rectum are given?

Let $P(x, y)$ be a variable point on the parabola $y = 4x^2 + 1$. Let $Q(c, c)$ be the foot of the perpendicular drawn from $P$ to the line $y = x$. If $R(h, k)$ is the mid-point of $PQ$,then the locus of $R$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module