JEE Main 2020 PYQ: the line y=mx+4 is tangent to both y²=4x and x²=2by. Find b by matching the tangency condition on the first parabola, then applying it on the second.
JEE Main 2019 PYQ: one end of a focal chord of the parabola y²=16x is at (1,4). Find the length of the chord by locating its other endpoint via substitution.
The normal at point P on the parabola y²=6x passes through (5,−8). If the tangent at P meets the directrix at Q, find the ordinate of Q using the parametric form and solving a cubic for the parameter.
The locus of the midpoint of the segment joining the focus of y²=4ax to a moving point on the parabola is another parabola. Find its directrix using the parametric form and midpoint formula.
P(4,4√3) lies on y²=4ax, and PQ is a focal chord. M, N are feet of perpendiculars from P, Q to the directrix. Find the area of quadrilateral PQMN using the focal-chord parameter relation.
Find point P on the parabola y²=8x at minimum distance from the centre C of x²+(y+6)²=1, then find the equation of the circle through C centred at P.
Find the shortest distance between the line y−x=1 and the parabola x=y² by parametrising a point on the curve and minimising the distance function.
The common tangents to 4(x²+y²)=9 and y²=4x meet at Q. An ellipse has semi-minor axis OQ and semi-major axis 6. Find l/e² for this ellipse.