JEE MainsJEE AdvanceBITSAT

JEE Mains | PYQ

Let \( P \) be the parabola, whose focus is \( (-2,1) \) and directrix is \( 2x + y + 2 = 0 \). Then the sum of the ordinates of the points on \( P \), whose abscissa is \( -2 \), is:

If the midpoint of a chord of the ellipse \( \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \) is \( \left(\sqrt{2}, \frac{4}{3}\right) \), and the length of the chord is \( \frac{2\sqrt{a}}{3} \), then \( \alpha \) is:

Let \( H_1 : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \) and \( H_2 : -\dfrac{x^2}{A^2} + \dfrac{y^2}{B^2} = 1 \) be two hyperbolas having length of latus rectums \( 15\sqrt{2} \) and \( 12\sqrt{5} \) respectively. Let their eccentricities be \( e_1 = \sqrt{\dfrac{5}{2}} \) and \( e_2 \) respectively. If the product of the lengths of their transverse axes is \( 100\sqrt{10} \), then \( 25e_2^2 \) is equal to \( \underline{\phantom{xxxx}} \).

Let the length of the transverse and conjugate axes of a hyperbola in standard form be \( 2a \) and \( 2b \) respectively, and one focus and the corresponding directrix of this hyperbola be \( ( +5,0 ) \) and \( 5x+9=0 \), respectively. If the product of the focal distances of a point \( (u,2\sqrt{5}) \) on the hyperbola is \( p \), then \( 4p \) equal to __________.

Let \( e_1 \) and \( e_2 \) be the eccentricities of the ellipse \( \dfrac{x^{2}}{b^{2}} + \dfrac{y^{2}}{25} = 1 \) and the hyperbola \( \dfrac{x^{2}}{16} - \dfrac{y^{2}}{b^{2}} = 1 \), respectively. If \( b < 5 \) and \( e_1 e_2 = 1 \), then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:

Let \( A \) and \( B \) be the two points of intersection of the line \( y + 5 = 0 \) and the mirror image of the parabola \( y^{2} = 4x \) with respect to the line \( x + y + 4 = 0 \). If \( d \) denotes the distance between \( A \) and \( B \), and \( a \) denotes the area of \( \Delta SAB \), where \( S \) is the focus of the parabola \( y^{2} = 4x \), then the value of \( (a + d) \) is ________.

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

The equation of the chord, of the ellipse \(\displaystyle \frac{x^2}{25} + \frac{y^2}{16} = 1 \), whose mid-point is \( (3, 1) \) is:

Let the length of a latus rectum of an ellipse \( \dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 \) be \( 10 \). If its eccentricity is the minimum value of the function \( f(t) = t^{2} + t + \dfrac{11}{12},\, t \in \mathbb{R} \), then \( a^{2} + b^{2} \) is equal to:

Let \( A=\{(\alpha,\beta)\in R\times R :|\alpha-1|\le 4 \text{ and } |\beta-5|\le 6\} \) and \( B=\{(\alpha,\beta)\in R\times R:\ 16(\alpha-2)^{2}+9(\beta-6)^{2}\le 144\} \). Then

If \( A \) and \( B \) are the points of intersection of circle \( x^{2}+y^{2}-8x=0 \) and the hyperbola \( \frac{x^{2}}{9}-\frac{y^{2}}{4}=1 \) and a point \( P \) moves on the line \( 2x-3y+4=0 \), then the centroid of \( \triangle PAB \) lies on the line:

Let \( y^{2}=12x \) be the parabola and \( S \) be its focus. Let \( PQ \) be a focal chord of the parabola such that \( (SP)(SQ)=\dfrac{147}{4} \). Let \( C \) be the circle described taking \( PQ \) as diameter. If the equation of a circle \( C \) is \( 64x^{2}+64y^{2}-\alpha x-64\sqrt{3}\,y=\beta \), then \( \beta-\alpha \) is equal to _______.

If the equation of the parabola with vertex \( V\left(\frac{3}{2},\,3\right) \) and the directrix \( x+2y=0 \) is \( \alpha x^{2}+\beta y^{2}-\gamma xy-30x-60y+225=0 \), then \( \alpha+\beta+\gamma \) is equal to: