Ellipse PYQs - Last 5 Years

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

Let the ellipse \(E:{x^2} + 9{y^2} = 9\) 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. I...

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

Let the eccentricity of the ellipse \({x^2} + {a^2}{y^2} = 25{a^2}\) be b times the eccentricity of the hyperbola \({x^2} - {a^2}{y^2} = 5\), where a is the minimum distance between the curves y = ex and y = logex. Then $${a^2} + {1 \over {...

Let a line L pass through the point of intersection of the lines \(b x+10 y-8=0\) and \(2 x-3 y=0, \mathrm{~b} \in \mathbf{R}-\left\{\frac{4}{3}\right\}\). If the line \(\mathrm{L}\) also passes through the point \((1,1)\) and touches the c...

Let the tangents at the points \(\mathrm{P}\) and \(\mathrm{Q}\) on the ellipse \(\frac{x^{2}}{2}+\frac{y^{2}}{4}=1\) meet at the point \(R(\sqrt{2}, 2 \sqrt{2}-2)\). If \(\mathrm{S}\) is the focus of the ellipse on its negative major axis,...

Let the eccentricity of an ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\), \(a > b\), be \({1 \over 4}\). If this ellipse passes through the point \(\left( { - 4\sqrt {{2 \over 5}} ,3} \right)\), then \({a^2} + {b^2}\) is...

If the length of the latus rectum of the ellipse \(x^{2}+4 y^{2}+2 x+8 y-\lambda=0\) is 4 , and \(l\) is the length of its major axis, then \(\lambda+l\) is equal to ____________.

The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse \({x^2} + 2{y^2} = 4\) is an ellipse with eccentricity :

If m is the slope of a common tangent to the curves \({{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1\) and \({x^2} + {y^2} = 12\), then \(12{m^2}\) is equal to :

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

The line y = x + 1 meets the ellipse \({{{x^2}} \over 4} + {{{y^2}} \over 2} = 1\) at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)2 is equal to :

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

If two tangents drawn from a point (\(\alpha\), \(\beta\)) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x are such that the slope of one tangent is four times the other, then the value of (10\(\alpha\) + 5)2 + (16\(\beta\)2 + 5...

Let the maximum area of the triangle that can be inscribed in the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2\), having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to t...