Current Affairs JEE Main & Advanced

The point of intersection of normals at any two points \[P(at_{1}^{2},2a{{t}_{1}})\] and \[Q(at_{2}^{2},2a{{t}_{2}})\]on the parabola \[{{y}^{2}}=4ax\]is \[R\,[2a+a(t_{1}^{2}+t_{2}^{2}+{{t}_{1}}{{t}_{2}}),\ -a{{t}_{1}}{{t}_{2}}({{t}_{1}}+{{t}_{2}})]\]      

If the normal at the point \[P(at_{^{1}}^{2},2a{{t}_{1}})\] meets the parabola \[{{y}^{2}}=4ax\] again at \[(at_{2}^{2},2a{{t}_{2}})\],         then \[{{t}_{2}}=-{{t}_{1}}-\frac{2}{{{t}_{1}}}\].  

The points on the curve at which the normals pass through a common point are called co-normal points. Q, R, S are co-normal points. The co- normal points are also called the feet of the normals.         Properties of co-normal points   (1) Three normals can be drawn from a point to a parabola.   (2) The algebraic sum of the slopes of three concurrent normals is zero.   (3) The sum of the ordinates of the co-normal points is zero.   (4) The centroid of the triangle formed by the co-normal points lies on the axis of the parabola.   (5) The centroid of a triangle formed by joining the foots of the normal of the parabola lies on its axis and is given by     \[\left( \frac{am_{1}^{2}+am_{2}^{2}+am_{3}^{2}}{3},\frac{2a{{m}_{1}}+2a{{m}_{2}}+2a{{m}_{3}}}{3} \right)\]\[=\left( \frac{am_{1}^{2}+am_{2}^{2}+am_{3}^{2}}{3},0 \right)\].     (6) If three normals drawn to any parabola \[{{y}^{2}}=4ax\]from a given point (h, k) be real, then \[h>2a\] for \[a=1\], normals drawn to the parabola \[{{y}^{2}}=4x\] from any point (h, k) are real, if \[h>2\].     (7) Out of these three at least one is real, as imaginary normals will always occur in pairs.

Let PQ and PR be tangents to the parabola \[{{y}^{2}}=4ax\] drawn from any external point \[P({{x}_{1}},{{y}_{1}})\] then QR is called the chord of contact of the parabola \[{{y}^{2}}=4ax\].         The chord of contact of tangents drawn from a point \[({{x}_{1}},{{y}_{1}})\] to the parabola \[{{y}^{2}}=4ax\] is \[y{{y}_{1}}=2a(x+{{x}_{1}})\].  

The equation of the chord at the parabola \[{{y}^{2}}=4ax\] bisected at the point \[({{x}_{1}},{{y}_{1}})\] is given by \[T={{S}_{1,}}\]               i.e.,  \[y{{y}_{1}}-2a(x+{{x}_{1}})\]\[y{{y}_{1}}-2a(x+{{x}_{1}})=y_{1}^{2}-4a{{x}_{1}}\]     where \[T=y{{y}_{1}}-2a(x+{{x}_{1}})\] and \[{{S}_{1}}=y_{1}^{2}-4a{{x}_{1}}\].  

Let \[P(at_{1}^{2},2a{{t}_{1}}),Q(at_{2,}^{2},2a{{t}_{2}})\]be any two points on the parabola \[{{y}^{2}}=4ax\]. Then, the equation of the chord joining these points is, \[y-2a{{t}_{1}}=\frac{2}{{{t}_{1}}+{{t}_{2}}}\left( x-at_{1}^{2} \right)\] or \[y({{t}_{1}}+{{t}_{2}})=2x+2a{{t}_{1}}{{t}_{2}}\].     (1) Condition for the chord joining points having parameters \[{{\mathbf{t}}_{\mathbf{1}}}\] and \[{{\mathbf{t}}_{\mathbf{2}}}\] to be a focal chord: If the chord joining points \[(at_{1}^{2},2a{{t}_{1}})\] and \[(at_{2}^{2},2a{{t}_{2}})\] on the parabola passes through its focus, then \[(a,0)\] satisfies the equation \[y({{t}_{1}}+{{t}_{2}})=2x+2a{{t}_{1}}{{t}_{2}}\]     \[\Rightarrow \] \[0=2a+2a{{t}_{1}}{{t}_{2}}\] \[\Rightarrow \] \[{{t}_{1}}{{t}_{2}}=-1\] or \[{{t}_{2}}=-\frac{1}{{{t}_{1}}}\].     (2) Length of the focal chord: The length of a focal chord having parameters \[{{t}_{1}}\]and \[{{t}_{2}}\]for its end points is \[a{{({{t}_{2}}-{{t}_{1}})}^{2}}\].

The locus of the middle points of a system of parallel chords is called a diameter and in case of a parabola this diameter is shown to be a straight line which is parallel to the axis of the parabola.             The equation of the diameter bisecting chords of the parabola \[{{y}^{2}}=4ax\]of slope \[m\] is \[y=2a/m\].  

Let the parabola \[{{y}^{2}}=4ax\]. Let the tangent and normal at \[P({{x}_{1}},{{y}_{1}})\] meet the axis of parabola at T and G respectively, and tangent at \[P({{x}_{1}},{{y}_{1}})\] makes angle \[\psi \] with the positive direction of x-axis. \[A(0,\,0)\] is the vertex of the parabola and \[PN=y\]. Then,           (1) Length of tangent \[=PT=PN\,\text{cosec}\,\,\psi ={{y}_{1}}\,\text{cosec}\,\psi \]   (2) Length of normal \[=PG=PN\text{cosec}({{90}^{o}}-\psi )={{y}_{1}}\sec \psi \]   (3) Length of subtangent \[=TN=PN\cot \psi ={{y}_{1}}\cot \psi \]   (4) Length of subnormal \[=NG=PN\cot ({{90}^{o}}-\psi )={{y}_{1}}\tan \psi \]   where, \[\tan \psi =\frac{2a}{{{y}_{1}}}=m\],  [Slope of tangent at \[P(x,\,\,y)\]]  

(1) Length of tangent at \[(a{{t}^{2}},2at)\]\[=2at\,\text{cosec}\psi \]      \[=2at\sqrt{(1+{{\cot }^{2}}\psi )}\] \[=2at\sqrt{1+{{t}^{2}}}\]     (2) Length of normal at \[(a{{t}^{2}},\,2at)=2at\sec \psi \]     \[=2at\sqrt{(1+{{\tan }^{2}}\psi )}\]\[=2a\sqrt{{{t}^{2}}+{{t}^{2}}{{\tan }^{2}}\psi }\] \[=2a\sqrt{({{t}^{2}}+1)}\]     (3) Length of subtangent at \[(a{{t}^{2}},2at)=2at\cot \psi \]\[=2a{{t}^{2}}\]     (4) Length of subnormal at \[(a{{t}^{2}},2at)=2at\tan \psi \]\[=2a\].

The locus of the point of intersection of the tangents to the parabola at the ends of a chord drawn from a fixed point P is called the polar of point P and the point P is called the pole of the polar.   Equation of polar : Equation of polar of the point \[({{x}_{1}},\,{{y}_{1}})\] with respect to parabola \[{{y}^{2}}=4ax\] is same as chord of contact and is given by \[y{{y}_{1}}=2a(x+{{x}_{1}})\].          Coordinates of pole : The pole of the line \[lx+my+n=0\] with respect to the parabola \[{{y}^{2}}=4ax\]is \[\left( \frac{n}{l},\frac{-2am}{l} \right)\].