HARD
JEE Main
IMPORTANT
Earn 100

Show that all chords of a parabola which subtend a right angle at the vertex pass through a fixed point on the axis of the curve.

Important Questions on Conic Section

HARD
JEE Main
IMPORTANT
Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 31

Show that if r1 and r2 be the lengths of perpendicular chords of a parabola drawn through the vertex, then

r1r243=16a2r123+r223

HARD
JEE Main
IMPORTANT
Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 32
Show that the equation of the circle described on the chord intercepted by the parabola y2=4ax on the line y=mx+c as diameter is

m2x2+y2+2xmc-2a-4amy+4amc+c2=0.

HARD
JEE Main
IMPORTANT
Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 33
The tangents to the parabola y2=4ax meet on the parabola y2=ax-2a. Prove that the normals at the points of contact intersect on the tangent at the vertex to the first parabola.
HARD
JEE Main
IMPORTANT
Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 34
Tangent is drawn at any point x1, y1 on the parabola y2=4ax. Now tangents are drawn from any point on this tangent to the circle x2+y2=a2 such that all the chords of contact pass through a fixed point x2, y2. Prove that y1y22=-4x1x2
HARD
JEE Main
IMPORTANT
Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 35
Find the points on the x-axis from which exactly three distinct chords (secants) of the circle x2+y2=a2 can be drawn which are bisected by the parabola y2=4ax, a>0.
HARD
JEE Main
IMPORTANT
Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 36
Prove that on the axis of any parabola, there is a certain point P which has the property that if a chord AB of the parabola, be drawn through it, then 1AP2+1BP2 is the same for all positions of the chord.
HARD
JEE Main
IMPORTANT
Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 37
Let P be a point on the parabola y2=4x, with the ordinate y satisfying 1<y≤2. The normal to the parabola at P intersects the x-axis in N and a line parallel to y-axis through P intersects the x-axis in M. If S be the focus of the parabola and z= area of ΔPMN- area of ΔPMS, find the maximum value of z.
HARD
JEE Main
IMPORTANT
Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 38
If a chord of the parabola y2=4ax touches the parabola y2=4bx, then show that the tangent at the extremities of the chord meet on the parabola by2=4a2x .