Number of p oints which can be expressed in form (p₁/q

English

Gujarati

Hindi

10-2. Parabola, Ellipse, Hyperbola

normal

The number of $p$ oints which can be expressed in the form $(p_1/q_ 1 , p_2/q_2)$, ($p_i$ and $q_i$ $(i = 1,2)$ are co-primes) and lie on the ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$ is

$4$

$8$

$12$

more than $12$

Solution

$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$

$\mathrm{P}(3 \cos \theta, 2 \sin \theta)$

$\left(\frac{3\left(1-\tan ^{2} \frac{\theta}{2}\right)}{1+\tan ^{2} \frac{\theta}{2}}, 2 \times \frac{2 \tan \frac{\theta}{2}}{1+\tan ^{2} \frac{\theta}{2}}\right)$

$\tan \frac{\theta}{2}$ can take infinite rational values.

Standard 11

Mathematics

The equation of normal at the point $(0, 3)$ of the ellipse $9{x^2} + 5{y^2} = 45$ is

medium

Let the ellipse $E : x ^2+9 y ^2=9$ intersect the positive $x$ – and $y$-axes at the points $A$ and $B$ respectively Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle which vertices $A, P$ and the origin $O$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m – n$ is equal to

hard

(JEE MAIN-2023)

How many real tangents can be drawn to the ellipse $5x^2 + 9y^2 = 32$ from the point $(2,3)$