The tangent at point (a cos theta, b sin thet | vedclass

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

Question

(B) The equation of the tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at the point $(a \cos 	heta, b \sin 	heta)$ is given by $\fra

Vedclass · Accepted Answer

$2ab$

Vedclass · Answer

(B) The equation of the tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at the point $(a \cos 	heta, b \sin 	heta)$ is given by $\frac{x \cos 	heta}{a} + \frac{y \sin 	heta}{b} = 1$.To find the $x$-intercept $T$, set $y = 0$: $\frac{x \cos 	heta}{a} = 1 \implies x = \frac{a}{\cos 	heta}$. Thus, $OT = \frac{a}{\cos 	heta}$.To find the $y$-intercept $T_1$, set $x = 0$: $\frac{y \sin 	heta}{b} = 1 \implies y = \frac{b}{\sin 	heta}$. Thus, $OT_1 = \frac{b}{\sin 	heta}$.The product $(OT)(OT_1) = \frac{a}{\cos 	heta} \cdot \frac{b}{\sin 	heta} = \frac{ab}{\sin 	heta \cos 	heta} = \frac{2ab}{\sin 2	heta}$.Since $0 Therefore, the minimum value is $2ab$.

The tangent at point $(a \cos \theta, b \sin \theta)$, where $0 < \theta < \frac{\pi}{2}$, to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ meets the $x$-axis at $T$ and the $y$-axis at $T_1$. Then the value of $\min_{0 < \theta < \frac{\pi}{2}} (OT)(OT_1)$ is

Similar Questions

If the lines joining the foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b$) to an extremity of its minor axis are inclined at an angle of $60^{\circ}$ to each other, then the eccentricity of the ellipse is:

$A$ line of fixed length $a + b$, where $a \neq b$, moves such that its ends are always on two fixed perpendicular straight lines. The locus of a point which divides the line into two parts of lengths $a$ and $b$ is

If tangents are drawn to the ellipse $x^2+2y^2=2$,then the locus of the midpoints of the intercepts made by the tangents between the coordinate axes is

If $A_1, A_2, A_3$ are the areas of the ellipse $x^2+4y^2-4=0$, its director circle, and its auxiliary circle respectively, then $A_2+A_3-A_1=$ (in $\pi$)

$A$ particle is travelling in a clockwise direction on the ellipse $\frac{x^2}{100} + \frac{y^2}{25} = 1$. If the particle leaves the ellipse at the point $(-8, 3)$ and travels along the tangent to the ellipse at that point,then the point where the particle crosses the $Y$-axis is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module