If α x+β y=109 is equation of chord of ellipse frac{x^2}{9}+frac{y^2}{4}=1 , whose mid point is le

English

Gujarati

Hindi

10-2. Parabola, Ellipse, Hyperbola

hard

If $\alpha x+\beta y=109$ is the equation of the chord of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, whose mid point is $\left(\frac{5}{2}, \frac{1}{2}\right)$, then $\alpha+\beta$ is equal to

A$37$

B$46$

C$58$

D$72$

(JEE MAIN-2025)

Solution

image
Equation of chord $T = S _1$
$\frac{5}{2}\left(\frac{x}{9}\right)+\frac{1}{2}\left(\frac{y}{4}\right)=\frac{25}{36}+\frac{1}{16}$
$\Rightarrow \frac{5 x}{18}+\frac{y}{8}=\frac{100+9}{144}=\frac{109}{144}$
$\Rightarrow 40 x+18 y=109$
$\Rightarrow \alpha=40, \beta=18$
$\Rightarrow \alpha+\beta=58$

Standard 11

Mathematics

Find the equation for the ellipse that satisfies the given conditions: Ends of major axis $(±3,\,0)$ ends of minor axis $(0,\,±2)$

medium

The locus of point of intersection of two perpendicular tangent of the ellipse $\frac{{{x^2}}}{{{9}}} + \frac{{{y^2}}}{{{4}}} = 1$ is :-

normal

For $0 < \theta < \frac{\pi}{2}$, four tangents are drawn at the four points $(\pm 3 \cos \theta, \pm 2 \sin \theta)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. If $A(\theta)$ denotes the area of the quadrilateral formed by these four tangents, the minimum value of $A(\theta)$ is

normal

(KVPY-2018)

The eccentricity of the ellipse $4{x^2} + 9{y^2} + 8x + 36y + 4 = 0$ is

medium

Identify the statements which are True.

normal

If $\alpha x+\beta y=109$ is the equation of the chord of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, whose mid point is $\left(\frac{5}{2}, \frac{1}{2}\right)$, then $\alpha+\beta$ is equal to

Solution

Similar Questions

Find the equation for the ellipse that satisfies the given conditions: Ends of major axis $(±3,\,0)$ ends of minor axis $(0,\,±2)$

The locus of point of intersection of two perpendicular tangent of the ellipse $\frac{{{x^2}}}{{{9}}} + \frac{{{y^2}}}{{{4}}} = 1$ is :-

For $0 < \theta < \frac{\pi}{2}$, four tangents are drawn at the four points $(\pm 3 \cos \theta, \pm 2 \sin \theta)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. If $A(\theta)$ denotes the area of the quadrilateral formed by these four tangents, the minimum value of $A(\theta)$ is

The eccentricity of the ellipse $4{x^2} + 9{y^2} + 8x + 36y + 4 = 0$ is

Identify the statements which are True.

Start a Free Trial Now