A tangent having slope m = -1/2 to the ellips | vedclass

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

Question

(A) The equation of the ellipse is $3x^2 + 4y^2 = 12$, which can be written as $\frac{x^2}{4} + \frac{y^2}{3} = 1$. Here $a^2 = 4$ and $b^2 = 3$.The e

Vedclass · Accepted Answer

$4 	ext{ sq. units}$

Vedclass · Answer

(A) The equation of the ellipse is $3x^2 + 4y^2 = 12$, which can be written as $\frac{x^2}{4} + \frac{y^2}{3} = 1$. Here $a^2 = 4$ and $b^2 = 3$.The equation of a tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with slope $m$ is $y = mx \pm \sqrt{a^2m^2 + b^2}$.Given $m = -1/2$, the equation becomes $y = -\frac{1}{2}x \pm \sqrt{4(-\frac{1}{2})^2 + 3} = -\frac{1}{2}x \pm \sqrt{1 + 3} = -\frac{1}{2}x \pm 2$.For the $X$-intercept $(A)$, set $y = 0$: $0 = -\frac{1}{2}x \pm 2 \implies x = \pm 4$. So $A = (\pm 4, 0)$.For the $Y$-intercept $(B)$, set $x = 0$: $y = \pm 2$. So $B = (0, \pm 2)$.The area of $\Delta AOB = \frac{1}{2} 	imes |x_A| 	imes |y_B| = \frac{1}{2} 	imes 4 	imes 2 = 4 	ext{ sq. units}$.

$A$ tangent having slope $m = -1/2$ to the ellipse $3x^2 + 4y^2 = 12$ intersects the $X$-axis and $Y$-axis at the points $A$ and $B$ respectively. If $O$ is the origin, then the area of $\Delta AOB$ is ...

Similar Questions

Let $e$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. If $a=5, b=4$ and the equation of the normal drawn at one end of the latus rectum that lies in the first quadrant is $lx+my=27$,then $l+m=$

The angle between the pair of tangents drawn from the point $(1, 2)$ to the ellipse $3x^2 + 2y^2 = 5$ is

$A$ point moves such that the sum of its distances from $(ae, 0)$ and $(-ae, 0)$ is $2a$. Then the equation to its locus,where $b^2 = a^2(1 - e^2)$,is

The equation of the ellipse whose axes are the coordinate axes,which passes through the point $(-3, 1)$,and has an eccentricity $e = \sqrt{\frac{2}{5}}$ is:

Find the equation of the ellipse whose major axis is $8$ and eccentricity is $1/2$ $(a > b)$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module