JEE Main Ellipse Practice Questions With Solutions

If the radius of the largest circle with centre (2,0) inscribed in the ellipse $x^2+4y^2=36$x^{2} + 4 y^{2} = 36 is r, then 12r${}^2$^{2} is equal to :

Consider ellipses ${\mathrm{E}}_k:k{x}^2+k^2{y}^2=1,k=1,2,\dots ,20$E_{k} : k x^{2} + k^{2} y^{2} = 1 , k = 1 , 2 , \ldots , 20. Let ${\mathrm{C}}_k$C_{k} be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse ${\mathrm{E}}_k$E_{k}. If $r_k$r_{k} is the radius of the circle ${\mathrm{C}}_k$C_{k}, then the value of $\sum _{k=1}^{20}\frac{1}{r_k^2}$\sum_{k = 1}^{20} \frac{1}{r_{k}^{2}} is :

Let a circle of radius 4 be concentric to the ellipse $15x^2+19y^2=285$15 x^{2} + 19 y^{2} = 285. Then the common tangents are inclined to the minor axis of the ellipse at the angle :

Let the ellipse $E:x^2+9y^2=9$E : x^{2} + 9 y^{2} = 9 intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle with vertices A, P and the origin O is $\frac{m}{n}$\frac{m}{n}, where m and n are coprime, then $m-n$m - n is equal to :

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $\alpha$\alpha and the number of persons who speak only Hindi is $\beta$\beta, then the eccentricity of the ellipse $25({\beta }^2{x}^2+{\alpha }^2{y}^2)={\alpha }^2{\beta }^2$25 \left(\right. \beta^{2} x^{2} + \alpha^{2} y^{2} \left.\right) = \alpha^{2} \beta^{2} is :

If the maximum distance of normal to the ellipse $\frac{x^2}{4}+\frac{y^2}{b^2}=1,b<2$\frac{x^{2}}{4} + \frac{y^{2}}{b^{2}} = 1 , b < 2, from the origin is 1, then the eccentricity of the ellipse is :

Let a line L pass through the point of intersection of the lines $bx+10y-8=0$b x + 10 y - 8 = 0 and $2x-3y=0,\mathrm{b}\in \mathbf{R}-\left\{\frac{4}{3}\right\}$2 x - 3 y = 0 , \textrm{ } b \in \mathbf{R} - \left{\right. \frac{4}{3} \left.\right}. If the line $\mathrm{L}$L also passes through the point $(1,1)$\left(\right. 1 , 1 \left.\right) and touches the circle $17(x^2+y^2)=16$17 \left(\right. x^{2} + y^{2} \left.\right) = 16, then the eccentricity of the ellipse $\frac{x^2}{5}+\frac{y^2}{{\mathrm{b}}^2}=1$\frac{x^{2}}{5} + \frac{y^{2}}{\left(\textrm{ } b\right)^{2}} = 1 is :

The acute angle between the pair of tangents drawn to the ellipse $2x^2+3y^2=5$2 x^{2} + 3 y^{2} = 5 from the point $(1,3)$\left(\right. 1 , 3 \left.\right) is :

If the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 meets the line $\frac{x}{7}+\frac{y}{2\sqrt{6}}=1$\frac{x}{7} + \frac{y}{2 \sqrt{6}} = 1 on the $x$x-axis and the line $\frac{x}{7}-\frac{y}{2\sqrt{6}}=1$\frac{x}{7} - \frac{y}{2 \sqrt{6}} = 1 on the $y$y-axis, then the eccentricity of the ellipse is :

Let the eccentricity of the ellipse $x^2+a^2{y}^2=25a^2$x^{2} + a^{2} y^{2} = 25 a^{2} be b times the eccentricity of the hyperbola $x^2-a^2{y}^2=5$x^{2} - a^{2} y^{2} = 5, where a is the minimum distance between the curves y = e^x and y = log_ex. Then $a^2+\frac{1}{b^2}$a^{2} + \frac{1}{b^{2}} is equal to :