Major Axis of Ellipse

Q. The equation of tangent at a point with eccentric angle $\theta$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

1. $\frac{x}{a}\sec \theta -\frac{y}{b}\tan \theta =1$
2. $\frac{x}{a}\cot \theta +\frac{y}{b}\sec \theta =1$
3. $\frac{x}{a}\tan \theta -\frac{y}{b}\sec \theta =1$

Q.

One focus of an Ellipse is (1, 0) with centre (0, 0). If the length of major axis is 6, its e =

1. $\frac{1}{4}$

2. $\frac{2}{3}$

3. $\frac{1}{3}$

4. $\frac{1}{2}$

Q. Let $S,S^{\prime }$ be the foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ whose eccentricity is '$e$'. $P$ is a variable point on the ellipse. Consider the locus of the incentre of the $△PS{S}^{\prime }$

Maximum area of rectangle inscribed in the locus is

1. $\frac{2ab{e}^2}{1+e}$
2. $\frac{2abe}{1-e}$
3. $\frac{abe}{1+e}$
4. None of these

Q. The length of major axis of the ellipse $(5x-10{)}^2+(5y+15{)}^2=\frac{(3x-4y+7{)}^2}{4}$ is

1. $10$
2. $\frac{20}{3}$
3. $\frac{20}{7}$
4. $4$

Q. $\mathrm{The}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{focal}\mathrm{radii}\mathrm{drawn}\mathrm{from}\mathrm{the}\mathrm{point}$ $(4\sqrt{3},5)\mathrm{on}\mathrm{the}\mathrm{ellipse}25x^2+16y^2=1600$ $\mathrm{is}$ $\mathrm{units}$.

1. 16
2. 20
3. 7
4. 13

Q. A circle and an ellipse have centres at $(0,0)$ and the circle passes through foci $F_1$ and $F_2$ of the ellipse, such that the two curves intersect at four points. Let $P$ be any one of their points of intersection. If the length of major axis of the ellipse is $17$ and area of the triangle $P{F}_1{F}_2$ is $30$, then distance between foci is

1. $11$
2. $12$
3. $13$
4. $15$