Q. A double ordinate of the parabola $y^2 = 4ax$ is of length 8a. It subtends an angle at the vertex equal to

Conic Sections Report Error

A

$ \frac{\pi}{2}$

41%

B

$ \frac{\pi}{4}$

24%

C

$ \frac{\pi}{6}$

24%

D

$ \frac{2\pi}{3}$

12%

Solution:

Let A be the vertex of the given parabola $y^2 = 4ax$.
PNQ is the double ordinate of length $8a$.
$\therefore PN = NQ = 4a$
Now the y co-ordinates of P and Q are 4a and -4a.
Therefore from the equation of the parabola $y^{2} = 4ax$, we get
$\left(4a\right)^{2} = 4ax$, or $x = 4a.$
$\therefore $ the co-ordinates of P and Q are $\left(4a, \,4a\right)$ and $\left(4a,\, -4a\right)$.
i.e. $AN = PN = NQ$
Thus, $∠NAP = 45^{\circ}, ∠NAQ = 45^{\circ}$
i.e $∠PAQ = \frac{\pi}{2}.$

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Q. A double ordinate of the parabola $y^2 = 4ax$ is of length 8a. It subtends an angle at the vertex equal to

$ \frac{\pi}{2}$

$ \frac{\pi}{4}$

$ \frac{\pi}{6}$

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1. Point P is moving along the circumference of the circle $(x - 4)^2 (y - 8)^2 = 16$. Then, it breaks away from it and moving along a tangent to the circle, cuts the x-axis at the point (8, 0). The coordinates of the point on the circle at which the moving point broke away could be

2. Latus rectum of the parabola whose focus is $(3,4)$ and whose tangent at vertex has the equation $x+y=7+5 \sqrt{2}$ is -

3. Directrix of a parabola is $x+y=2$. If it's focus is origin, then latus rectum of the parabola is equal to -

4. Which one of the following equations represents parametrically, parabolic profile?

5. Let $C$ be a circle and $L$ a line on the same plane such that $C$ and $L$ do not intersect. Let $P$ be a moving point such that the circle drawn with centre at $P$ to touch $L$ also touches $C$. Then the locus of $P$ is -

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4. If $\begin{bmatrix}1&- \tan\theta \\ \tan \theta&1\end{bmatrix}\begin{bmatrix}1&\tan \theta \\ - \tan \theta &1\end{bmatrix}^{-1} = \begin{bmatrix}a&-b\\ b&a\end{bmatrix}$ then

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