ParabolaMedium
Question
Let O be the vertex of the parabola $y^{2} = 4x$ and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C , then the length of its latus rectum is :
Options
A.1
B.2
C.4
D.8
Solution
Sol. [Key Concept: For $y^{2} = 4x$ with perpendicular chords $OP,OQ$: parametrise $P(t_{1}^{2},2t_{1}),Q(t_{2}^{2},2t_{2})$ with $t_{1}t_{2} = - 4$; the midpoint locus is a parabola of latus rectum $2$.]
$${t_{1}t_{2} = - 4 }{2\text{ }h = t_{1}\ ^{2} + t_{2}\ ^{2}\ 2k = 2\left( t_{1} + t_{2} \right) }{2\text{ }h = k^{2} - 2t_{1}t_{2} }{2\text{ }h = k^{2} + 8 }{k^{2} = 2h - 8 }{= 2((\text{ }h - 4) }$$L.R. $= 2$
Create a free account to view solution
View Solution FreeMore Parabola Questions
The curve described parametrically by x = t2 + t +1, y = t2 - t +1 represents...PQ is a normal chord of the parabola y2 = 4ax at P, A being the vertex of the parabola. Through P a line is drawn parall...Let $y^{2} = 4ax$ be a parabola and $x^{2} - y^{2} = a^{2}$ be a hyperbola. Then number of common tangents is :...Length of the normal chord of the parabola, y2 = 4x, which makes an angle of with the axis of x is -...The distance between a tangent to the parabola $y^{2} = 4Ax(A > 0)$ and the parallel normal with gradient 1 is :...