Polar Line and Pole of a Conic
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Direct answer
The polar of a point P(x1, y1) with respect to a conic is the straight line T = 0 built by halving the mixed terms: for the circle x^2 + y^2 = a^2 it is xx1 + yy1 = a^2; for the general circle x^2 + y^2 + 2gx + 2fy + c = 0 it is xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0; for the parabola y^2 = 4ax it is yy1 = 2a(x + x1); for the ellipse x^2/a^2 + y^2/b^2 = 1 it is xx1/a^2 + yy1/b^2 = 1. When P lies on the conic, its polar is the tangent at P; when P lies outside, the polar is the chord of contact of the two tangents from P. The relationship is reciprocal — La Hire's theorem: P lies on the polar of Q exactly when Q lies on the polar of P — and for the parabola the polar of the focus is the directrix itself.
What you must remember
- Circle polar: xx1 + yy1 = a^2 for x^2 + y^2 = a^2; with general equation use the T-form with g(x + x1) + f(y + y1) + c.
- Parabola polar: yy1 = 2a(x + x1) for y^2 = 4ax; hyperbola and ellipse follow the same xx1/a^2 ± yy1/b^2 = 1 pattern.
- Tangent as special case: P on the conic makes T = 0 the tangent at P — the same formula serves tangents and polars, which is why memorising one T = 0 covers both chapters.
- Chord of contact: P outside the conic gives two real tangents touching at Q and R; the polar is the line QR.
- La Hire's theorem: P on the polar of Q ⟺ Q on the polar of P — the pole-polar relation is symmetric.
- Conjugate points and lines: two points are conjugate when each lies on the other's polar; two lines are conjugate when each passes through the other's pole; the centre of a central conic has the line at infinity as its polar.
- Focus-directrix cameo: for the parabola y^2 = 4ax, the polar of the focus (a, 0) is y·0 = 2a(x + a), i.e., x = -a, the directrix; for central conics, the polar of a focus is the corresponding directrix.
Finding and using one polar
Take the circle x^2 + y^2 = 9 and the exterior point P(4, 5). Its polar is 4x + 5y = 9. Where does this line sit? Its distance from the centre is 9/√(16 + 25) = 9/√41 ≈ 1.4, well inside the radius 3, so the polar cuts the circle in two real points — and those points are exactly the contact points of the two tangents drawn from P (the chord of contact). The geometry is self-checking: an exterior point has its polar pass inside, and an interior point has its polar fall wholly outside (distance a^2/d exceeds the radius when d < a). Now exercise La Hire: pick Q(0, 9/5), which satisfies 4 × 0 + 5 × (9/5) = 9, so Q lies on the polar of P. The polar of Q is 0·x + (9/5)y = 9, that is, y = 5 — and P(4, 5) lies on it, as reciprocity demands. Two computations, one theorem confirmed, and the pattern generalises verbatim to ellipse and parabola with their T = 0 forms.
How the exam frames it
JEE Main asks for the polar (chord of contact) of a given point with respect to a given circle or parabola — essentially the T = 0 formula evaluated — and for the pole of a given line by solving the coefficient matching. Advanced deploys La Hire and conjugacy: find the locus of points whose polars pass through a fixed point (answer: the polar of that point — a locus question answered by one line), or prove concyclicity of contact points. The predictable slips: using T = 0 with the point coordinates swapped into the wrong slots (the pattern is xx1 and yy1 — cross terms do not exist in these forms); treating the polar of an interior point as a chord of contact (no real tangents exist from inside, so the polar misses the conic); and confusing the polar with the normal or the diameter through the point. A favourite property-based multiple choice: the polars of all points on a straight line are concurrent — at the pole of that line — which is La Hire read backwards. Conic sections and tangents sit in both syllabi, with pole-polar among the classic Advanced extensions.
Frequently asked questions
What is the polar of a point with respect to x^2 + y^2 = a^2?
The line xx1 + yy1 = a^2; for the general circle, xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0.
When does the polar become the chord of contact?
When the point lies outside the conic: the two real tangents from it touch at two points, and the polar is the line joining them.
What does La Hire's theorem say?
Point P lies on the polar of Q if and only if Q lies on the polar of P — the pole-polar relation is perfectly symmetric.
What is the polar of a point lying on the conic itself?
The tangent at that point — the single formula T = 0 degenerates from chord of contact to tangent.
What is the polar of the focus of y^2 = 4ax?
The directrix x = -a; more generally each focus of a conic has the corresponding directrix as its polar.