Current Affairs JEE Main & Advanced

(1) Chord of contact : The chord joining the points of contact of the two tangents to a conic drawn from a given point, outside it, is called the chord of contact of tangents.       (2) Equation of chord of contact : The equation of the chord of contact of tangents drawn from a point \[({{x}_{1}},\,{{y}_{1}})\] to the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] is \[x{{x}_{1}}+y{{y}_{1}}={{a}^{2}}.\]     Equation of chord of contact at \[({{x}_{1}},\,{{y}_{1}})\] to the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] is \[x{{x}_{1}}+y{{y}_{1}}+g(x+{{x}_{1}})+f(y+{{y}_{1}})+c=0\].     It is clear from above that the equation to the chord of contact coincides with the equation of the tangent, if point \[({{x}_{1}},\,{{y}_{1}})\] lies on the circle.     The length of chord of contact \[=2\sqrt{{{r}^{2}}-{{p}^{2}}}\]; (p being length of perpendicular from centre to the chord)     Area of \[\Delta APQ\] is given by \[\frac{a{{(x_{1}^{2}+y_{1}^{2}-{{a}^{2}})}^{3/2}}}{x_{1}^{2}+y_{1}^{2}}\].     (3) Equation of the chord bisected at a given point : The equation of the chord of the circle \[S\equiv {{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] bisected at the point \[({{x}_{1}},\,{{y}_{1}})\] is given by  \[T={{S}_{1}}\].     i.e., \[x{{x}_{1}}+y{{y}_{1}}+g(x+{{x}_{1}})+f(y+{{y}_{1}})+c=x_{1}^{2}+y_{1}^{2}+2g{{x}_{1}}+2f{{y}_{1}}+c\].

  The normal of a circle at any point is a straight line, which is perpendicular to the tangent at the point and always passes through the centre of the circle.           (1) Equation of normal: The equation of normal to the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] at any point \[({{x}_{1}},\,{{y}_{1}})\] is \[y-{{y}_{1}}=\frac{{{y}_{1}}+f}{{{x}_{1}}+g}(x-{{x}_{1}})\] or \[\frac{x-{{x}_{1}}}{{{x}_{1}}+g}=\frac{y-{{y}_{1}}}{{{y}_{1}}+f}\].    The equation of normal to the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] at any point \[a{{x}^{2}}+2hxy+b{{y}^{2}}\] is \[x{{y}_{1}}-{{x}_{1}}y=0\] or \[\frac{x}{{{x}_{1}}}=\frac{y}{{{y}_{1}}}\].   (2) Parametric form : Since parametric co-ordinates of a point on the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] is \[+2gx+2fy+c=0\].   \[\therefore \] Equation of normal at\[(a\,\,\cos \theta ,\,\,\,a\,\,\sin \theta )\] is \[\frac{x}{a\,\cos \theta }=\frac{y}{a\,\sin \,\theta }\]    or  \[\frac{x}{\cos \theta }=\frac{y}{\sin \,\theta }\] or  \[y=x\,\,\tan \,\theta \]  or  \[y=mx\] where \[m=\tan \,\,\theta \], which is slope form of normal.

Let \[P({{x}_{1}},{{y}_{1}})\] be a point outside the circle and PAB and PCD drawn two secants. The power of \[P({{x}_{1}},{{y}_{1}})\] with respect to \[S\equiv {{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] is equal to PA. PB which is           \[x_{1}^{2}+y_{1}^{2}+2g{{x}_{1}}+2f{{y}_{1}}+c={{S}_{1}}\]   \[\therefore \,\,\,\,PA\,.\,PB={{(\sqrt{{{S}_{1}}})}^{2}}=\]     Square of the length of tangent.   If P is outside, inside or on the circle then PA. PB is \[+ve\], \[-ve\] or zero respectively.

The locus of the point of intersection of two perpendicular tangents to a circle is called the Director circle.     Let the circle be \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\], then equation of director circle is \[{{x}^{2}}+{{y}^{2}}=2{{a}^{2}}\].     Obviously director circle is a concentric circle whose radius is \[\sqrt{2}\] times the radius of the given circle. Director circle of circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] is \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+2c-{{g}^{2}}-{{f}^{2}}=0\].  

From a given point \[P({{x}_{1}},{{y}_{1}})\] two tangents PQ and PR can be drawn to the circle \[S={{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0.\]     Their combined equation is \[S{{S}_{1}}={{T}^{2}}\],         where \[S=0\] is the equation of circle, \[T=0\] is the equation of tangent at \[({{x}_{1}},\,{{y}_{1}})\] and  \[{{S}_{1}}\] is obtained by replacing  \[x\] by \[{{x}_{1}}\] and \[y\] by \[{{y}_{1}}\] in S.

Let \[PQ\] and \[PR\] be two tangents drawn from \[P({{x}_{1}},{{y}_{1}})\] to the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0.\]         Then \[PQ=PR\] is called the length of tangent drawn from point  \[P\] and is given by \[PQ=PR\] \[=\sqrt{x_{1}^{2}+y_{1}^{2}+2g{{x}_{1}}+2f{{y}_{1}}+c}=\sqrt{{{S}_{1}}}\].  

(1) Point form             (i) The equation of tangent at \[({{x}_{1}},{{y}_{1}})\] to circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] is \[x{{x}_{1}}+y{{y}_{1}}={{a}^{2}}\].     (ii) The equation of tangent at \[({{x}_{1}},{{y}_{1}})\] to circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]is\[x{{x}_{1}}+y{{y}_{1}}+g(x+{{x}_{1}})+f(y+{{y}_{1}})+c=0\].     (2) Parametric form : Since parametric co-ordinates of a point on the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] is \[(a\,\cos \theta ,\,a\,\sin \theta ),\] then equation of tangent at \[(a\,\cos \theta ,\,a\,\sin \theta )\] is \[x.\,a\,\cos \theta +y\,.\,a\,\sin \theta ={{a}^{2}}\]     or \[x\,\cos \theta +y\,\sin \theta =a\].     (3) Slope form : The straight line \[y=mx+c\] touches the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] if \[{{c}^{2}}={{a}^{2}}(1+{{m}^{2}})\] and the point of contact of tangent \[y=mx\pm a\sqrt{1+{{m}^{2}}}\] is \[\left( \frac{\mp ma}{\sqrt{1+{{m}^{2}}}},\,\,\frac{\pm a}{\sqrt{1+{{m}^{2}}}} \right)\].

The length of the intercept cut off from the line \[y=mx+c\] by the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] is \[2\sqrt{\frac{{{a}^{2}}(1+{{m}^{2}})-{{c}^{2}}}{1+{{m}^{2}}}}\].    (i) If \[{{a}^{2}}(1+{{m}^{2}})-{{c}^{2}}>0\], line will meet the circle at two real and different points.    (ii) If \[{{c}^{2}}={{a}^{2}}(1+{{m}^{2}})\], line will touch the circle.    (iii) If \[{{a}^{2}}(1+{{m}^{2}})-{{c}^{2}}<0\], line will meet the circle at two imaginary points.

A point \[({{x}_{1}},\,{{y}_{1}})\] lies outside, on or inside a circle \[S\equiv {{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] according as \[{{S}_{1}}\equiv x_{1}^{2}+y_{1}^{2}+2g{{x}_{1}}+2f{{y}_{1}}+c\] is positive, zero or negative.    The least and greatest distance of a point from a circle: Let \[S=0\] be a circle and \[A\,({{x}_{1}},\,{{y}_{1}})\] be a point. If the diameter of the circle through A is passing through the circle at P and Q, then \[AP=\ |AC-r|\ =\] least distance; \[AQ=AC+r=\] greatest distance where \['r'\] is the radius and C is the centre of the circle.        

The lengths of intercepts made by the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] on \[X\] and \[Y\] axes are \[2\sqrt{{{g}^{2}}-c}\] and \[2\sqrt{{{f}^{2}}-c}\] respectively.    Therefore,    (i) The circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] cuts the x-axis in real and distinct points, touches or does not meet in real points according as \[{{g}^{2}}>,=\,\ \text{or}\ \,<c\].    (ii) Similarly, the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] cuts the y-axis in real and distinct points, touches or does not meet in real points according as \[{{f}^{2}}>,=\,\ \text{or}\ \,<c\].