Q. Let P be 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 $\frac{tan \,∠PQF}{tan \,∠PFQ}$ is

WBJEEWBJEE 2013 Report Error

1 1/2

2 1/4

Solution:

Equation of parabola is
$y^{2}=4 a x ..... (i) $

Let the parametric coordinate of point $P$ on the parabola is
$(a, 2 a)$. Now, $Q F=2 \sqrt{2} a$
$P Q=2 a$ and $P F=2 a$
we observe that, $Q F^{2}=P Q^{2}+P F^{2}$
$\Rightarrow 8 a^{2}=4 a^{2}+4 a^{2}=8 a^{2}$
So, $\Delta Q P F$ form a right angle isoceles triangle.
In which, $\angle P Q F=\angle P F Q$
$\Rightarrow \tan \angle P Q F=\tan \angle P F Q$
$\Rightarrow \frac{\tan \angle P Q F}{\tan \angle P F Q}=1$

Questions from WBJEE 2013

1. The number of solutions of the equation $x+y+z = 10$ in positive integers $x,y,z$ is equal to

Binomial Theorem

2. For $0 \le P,Q \le \frac{\pi}{2}$, if sin P + cos $Q=2$, then the value of tan $\left(\frac{P + Q}{2}\right)$ is equal to

3. If $\alpha$ and $\beta$ are the roots of $x^2 - x+1 = 0$, then the value of $\alpha^{2013} + \beta^{2013}$ is equal to

Complex Numbers and Quadratic Equations

4. The value of the integral
$ \int\limits^{+1}_{ -1} \left\{\frac{x^{2013}}{e^{\left|x\right|} \left(x^{2} + cos\,x\right)}+\frac{1}{e^{\left|x\right|}}\right\}dx$
is equal to

Integrals

5. Let
$f(x) = 2^{100} x + 1$,
$g(x) = 3^{100} x + 1$.
Then the set of real numbers $x$ such that $f(g(x)) = x$ is

Relations and Functions - Part 2

Mathematics Most Viewed Questions

1. The solution of $\frac{dy}{dx} = \frac{y}{x}+\tan \frac{y}{x}$ is

WBJEE 2011 Differential Equations

2. The solution of the differential equation $\frac{dy}{dx} = (x +y)^2$ is

COMEDK 2009 Differential Equations

3. If a and b are vectors such that $|a+b|=|a-b|$ then the angle between a and b is

KCET 2007 Vector Algebra

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

COMEDK 2009 Matrices

5. $\int\frac{1}{\sin x\, \cos x}$ dx is equal to

KEAM 2016 Integrals

6. $\int\frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} dx $ is equal to :
(where $c$ is a constant of integration)

JEE Main 2019 Integrals

7. A particle is dropped under gravity from rest from a height $ h(g=9.8\,m/{{s}^{2}}) $ and it travels a distance $ \frac{9h}{25} $ in the last second the height $ h $ is:

Bihar CECE 2006

8. The value of $ \int{\frac{{{x}^{2}}+1}{{{x}^{4}}-{{x}^{2}}+1}}dx $ is

KEAM 2007 Integrals

9. The value of $\int \frac{x^2+1}{x^2-1}dx$ is

KCET 2007 Integrals

10. The value of the integral $\int\frac{cos x}{sin x + cos x}dx$ is equal to

KEAM 2013 Integrals