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JEE Advance | PYQ

Let \( P \) be the point on the parabola \( y=x^2 \) such that the slope of the tangent to the parabola at the point \( P \) is \( 4 \). Let \( Q \) be the point in the first quadrant lying on the circle \( x^2+y^2=2 \) such that the slope of the tangent to the circle at the point \( Q \) is \( -1 \). Let \( R \) be the point in the first quadrant lying on the ellipse \( x^2+4y^2=8 \) such that the slope of the tangent to the ellipse at the point \( R \) is \( -\frac{1}{2} \). Then the radius of the circle passing through the points \( P, Q \) and \( R \) is:

Consider the ellipses given by \( x^{2}+4y^{2}=1 \) and \( 4x^{2}+y^{2}=1 \).
Let \( P \) be the point in the first quadrant where the given ellipse intersect. If \( \theta \) is the acute angle between the tangents to the given ellipses at the point \( P \), then the value of \( 4\tan\theta \) is __________________.

Let \( T \) be the tangent to the parabola \( y^{2}=16x \) at the point \( (64,\,32) \). Let \( L \) be the tangent to the same parabola at another point \( (x_{1},y_{1}) \) on the parabola. If \( L \) and \( T \) are perpendicular to each other, then the distance between the point \( (x_{1},y_{1}) \) and the focus of the parabola, is:

Consider the ellipse \( E \) given by \( \frac{x^2}{18} + \frac{y^2}{12} = 1 \). Let \( H \) be the hyperbola whose eccentricity is the reciprocal of the eccentricity of \( E \) and whose foci are the same as that of \( E \). Let \( P \) and \( Q \) be the points of intersection of \( H \) and the parabola \( \sqrt{5}\,y = x^2 \) in the first quadrant. Let \( d \) be the distance between \( P \) and \( Q \).

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Consider the ellipses given by \( x^2 + 4y^2 = 1 \) and \( 4x^2 + y^2 = 1 \).
If \( \alpha \) is the area of the common region that lies inside both the given ellipses, then the value of \( \cot \alpha \) is ____________.