Let a tangent to Curve 9 x^2+16 y^2=144 intersect coordinate axes at points A and B . Then, minimum

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10-2. Parabola, Ellipse, Hyperbola

hard

Let a tangent to the Curve $9 x^2+16 y^2=144$ intersect the coordinate axes at the points $A$ and $B$. Then, the minimum length of the line segment $A B$ is $.........$

$5$

$6$

$7$

$8$

(JEE MAIN-2023)

Solution

Equation of tangent at point $P (4 \cos \theta, 3 \sin \theta)$ is $\frac{x \cos \theta}{4}+\frac{y \sin \theta}{3}=1$ So A is $(4 \sec \theta, 0)$ and point $B$ is $(0,3 \operatorname{cosec} \theta)$

Length $A B =\sqrt{16 \sec ^2 \theta+9 \operatorname{cosec}^2 \theta}$ $=\sqrt{25+16 \tan ^2 \theta+9 \cot ^2 \theta} \geq 7$

Standard 11

Mathematics

If $y = mx + c$ is tangent on the ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$, then the value of $c$ is

easy

If tangents are drawn from point $P(3\ sin\theta + 4\ cos\theta , 3\ cos\theta\ -\ 4\ sin\theta)$ , $\theta = \frac {\pi}{8}$ to the ellipse $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 1$ then angle between the tangents is

normal

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $36 x^{2}+4 y^{2}=144$

medium

The distance between the foci of an ellipse is 16 and eccentricity is $\frac{1}{2}$. Length of the major axis of the ellipse is

easy

Let $S=\left\{(x, y) \in N \times N : 9(x-3)^{2}+16(y-4)^{2} \leq 144\right\}$ and $ T=\left\{(x, y) \in R \times R :(x-7)^{2}+(y-4)^{2} \leq 36\right\}$ Then $n ( S \cap T )$ is equal to $……$