# Sections of a Cone, Circles

Tips:

• If we take the intersection of a plane with a cone, the section so obtained is called a conic section.
• When the plane cuts the double napped cone with different angles circle, ellipse, parabola and hyperbola are formed.
• ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a circle if h=0; a = b ≠ 0 and g2 + f2 – ac ≥ 0
• General equation of standard form of a circle is x2 + y2 + 2gx + 2fy + c = 0 if g2 + f2 – c > 0 then real circle, if g2 + f2 – c = 0 then point circle, if g2 + f2 – c < 0 then imaginary circle.
• The general equation of a circle is ax2 + ay2 + 2gx + 2fy + c = 0 its centre is \tt \left(\frac{-g}{a},\frac{-f}{a}\right) radius = \tt \frac{\sqrt{g^{2}+f^{2}-ac}}{\mid a\mid}
• The equation of the circle passing through origin and through the points of intersection of the line ax + by + c = 0 with co-ordinate axes is ab(x2 + y2) + c(bx + ay) = 0
• The equation of the circle with (x1 y1) and (x2 y2) a extremities of a diameter is (x - x1) (x - x2)+ (y – y1) (y – y2) = 0
• Any circle touching x-axis is of the form x2 + y2 +2gx + 2fy + g2 = 0. [c = g2] radius = |f|; if it is touching both axes then c = g2 = f2 and radius = |g| = |f|.
• Conditions for a line is tangent to a circle y = mx + c touches x2 + y2 = r2 is c2 = r2 (1 + m2)
• Suppose lx + my + n = 0 touches x2 + y2 = r2 then n2 = r2 (l2 + m2) and the point of contact is \tt \left(\frac{-lr^{2}}{n},\frac{-mr^{2}}{n}\right)
• Suppose lx + my + n = 0 touches x2 + y2 +2gx + 2fy + c = 0 is (l2 + m2) (g2 + f2 – c) = (lg + mf – n)2
• The equation of tangent to the circle x2 + y2 = r2 whose slope ‘m’ is \tt y = mx\ \pm\ r\sqrt{m^{2}+1}
• The intercept made by the circle x2 + y2 +2gx + 2fy + c = 0 on x-axis is \tt 2\sqrt{g^{2}-c}, on y-axis is \tt 2\sqrt{f^{2}-c}
• If 'r' is radius of a circle then a line which is at a distance ‘d’ from centre of the circle, cuts (a chord) an intercept of length \tt 2\sqrt{r^{2}-d^{2}}
• Equation of the tangent at ‘θ’ to the circle x2 + y2 = r2 is x cos θ + y sin θ = r
• Equation of the normal at ‘θ’ to the circle x2 + y2 = r2 is x sin θ – y cos θ = 0
• The equation of the normal at P(x1 y1) to x2 + y2 +2gx + 2fy + c = 0 is y1 (x + g) – x1 (y + f) = 0
• Equation of the chord joining the two points θ1 and θ2 of the circle x2 + y2 = r2 is \tt x \cos \left(\frac{\theta_{1}+\theta_{2}}{2}\right)+y \sin \left(\frac{\theta_{1}+\theta_{2}}{2}\right)= r \cos \left(\frac{\theta_{1}-\theta_{2}}{2}\right)
• The length of chord \tt \overline{AB} joining A(θ1), B(θ2) of the circle x2 + y2 = r2 (or) \tt \left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}= r^{2}\ is\ 2r \begin{vmatrix}\sin \frac{\left(\theta_{1}-\theta_{2}\right)}{2}\end{vmatrix}
• If 'C' is centre and ‘r’ is radius of a circle then the power of the point ‘P’ is defined as CP2 – r2
• Power of a point on the circle is zero, power is positive if point lies outside the circle power is negative if point lies inside the circle.
• If P(x1 y1) is a point lying outside of the circle S = 0 then the length of the tangent from ‘P’ is \tt \sqrt{S_{11}}
• If ‘P’ is a point (other than centre) in the plane of a circle and a secant line through ‘P’ cuts the circle in A and B. If the tangents at A and B intersect in Q, then the locus of ‘Q’ is a straight line called the polar of ‘P’ w.r.t the circle. ‘P’ is pole of the line.
• Pole of line lx + my + n = 0 w.r.t the circle x2 + y2 = r2 is \left(\frac{-lr^{2}}{n},\frac{-mr^{2}}{n}\right)
• The condition that the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate w.r.t the circle x2 + y2 = r2 is r2 (l1 l2 + m1 m2 ) = n1 n2
• The condition that the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate w.r.t the circle x2 + y2 + 2gx + 2fy + c = 0 is (g2 + f2 – c) (l1 l2 + m1 m2) = (l1 g + m1 f – n1) (l2 g + m2 f – n2)
• If 'c' is centre 'r' is radius of a circle and P, Q are a pair of inverse points then CP.CQ = r2
• The inverse point of (x1 y1) with respect to circle x2 + y2 = r2 is \tt \left[\frac{r^{2}x_{1}}{x_1^2+y_1^2}+\frac{r^{2}y_{1}}{x_1^2+y_1^2}\right]
• The inverse point of origin with respect to the circle x2 + y2 + 2gx + 2fy + c = 0 is \tt \left[\frac{-gc}{g^2+f^2}+\frac{-fc}{g^2+f^2}\right]
• Equation of the pair of tangents from origin to the circle x2 + y2 + 2gx + 2fy + c = 0 is (gx + fy)2 = c (x2 + y2)
• Area of the quadrilateral formed by the two tangents drawn from an external point to a circle and a pair of radii through their points of contact is \tt r\sqrt{S_{11}}
• The length of the chord of contact of the point P(x1 y1) w.r.t the circle S=0 is \tt 2r\sqrt{\frac{S_{11}}{S_{11}+r^{2}}}
• S=0 is a circle in standard form, with centre ‘c’ and radius r. If P(x1 y1) is a point then the area of the triangle formed by pair of tangents from ‘P’ and chord of contact of ‘P’ is \tt \frac {r{\left(S_{11}\right)^{3/2}}}{S_{11}+r^{2}}
• The locus of point of intersection of two tangents which include an angle ‘θ’ w.r.t the circle x2 + y2 = r2 is x2 + y2 = r2 cosec2 θ/2
• The locus of midpoints of chords of circle x2 + y2 = r2 which subtend angle ‘θ’ at centre is x2 + y2 = r2 cos2 θ/2
• Angle of intersection of ‘2’ circles is \tt \cos \theta\ = \frac{r_1^2+r_2^2-d^{2}}{2\ r_{1}r_{2}}
• Tricks
• The angle between the tangents from (α, β) to the circle x2 + y2 = a2 is \tt 2\ \tan^{-1}\left[\frac{a}{\sqrt{\alpha^{2}+\beta^{2}-a^{2}}}\right]
• If OA and OB are the tangents from the origin to the circle x2 + y2 + 2gx + 2fy + c = 0 and ‘c’ is the centre of the circle, then the area of the quadrilateral OACB is \tt \sqrt{c.\left(g^{2}+f^{2}-c\right)}
• The length of the common chord of the circles x2 + y2 + ax + by + c = 0 and x2 + y2 + bx + ay + c = 0 is \tt \sqrt{\frac{1}{2}\left(a+b\right)^{2}-4c}
• If 'O' is the origin and OP, OQ are tangents to the circle x2 + y2 + 2gx + 2fy + c = 0, then the circumcentre of the triangle OPQ is \tt \left(\frac{-g}{2},\frac{-f}{2}\right)
• The length of the chord intercepted by the circle x2 + y2 = r2 on the line \tt \frac{x}{a}+\frac{y}{b}=1\ is\ 2\sqrt{\left(\frac{r^{2}\left(a^{2}+b^{2}\right)-a^{2}b^{2}}{a^{2}+b^{2}}\right)}
• cos θ ∈ (-∝, -1) U (1, ∝) ⇒ circles do not intersect cos θ ∈ (-1, 1) ⇒ circles intersect each other cos θ = 0 ⇒ circles intersect each other orthogonally cos θ ∈ {-1, 1} ⇒ circles touch each other internally or externally

### Circle Examples

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1. The general equation of a circle is given by x2 + y2 + 2gx + 2fy + c = 0, where centre fo the circle = (−g, −f)
∴ Radius of the circle = \tt \sqrt{g^{2}+f^{2}-c}
2. Equation of circle having centre (h, k) and radius a is (x − h)2 + (y − k)2 = a2.
If centre is (0, 0), then equation of circle is x2 + y2 = a2.
3. When the circle passes through the origin, then equation of the circle is x2 + y2 − 2hx − 2ky = 0.

4. When the circle touches the X-axis, the equation is x2 + y2 − 2hx − 2ay + h2 = 0.

5. Equation of the circle, touching the Y-axis is x2 + y2 − 2ax − 2ky + k2 = 0.

6. Equation of the circle, touching both axes is x2 + y2 − 2ax − 2ay + a2 = 0.

7. Equation of the circle passing through the origin and centre lying on the X-axis is x2 + y2 − 2ax = 0.

8. Equation of the circle passing through the origin and centre lying on the Y-axis is x2 + y2 − 2ay = 0.

9. Equation of the circle through the origin and cutting intercepts a and b on the coordinate axes is x2 + y2 − by = 0.

10. Equation of the circle, when the coordinates of end points of a diameter are (x1 , y1) and (x2 , y2) is
(x − x1) (x − x2) + (y − y1) (y − y2) = 0.
11. Equation of the circle passes through three given points (x1 , y1) , (x2 , y2) and (x3 , y3) is \tt \begin{vmatrix}x^{2}+ y^{2} & x & y & 1 \\ x_1^2 + y_1^2 & x_1 & y_1 & 1 & \\ x_2^2 + y_2^2 & x_2 & y_2 & 1 \\ x_3^2 + y_3^2 & x_3 & y_3 & 1 \end{vmatrix}=0.
12. Parametric equation of a circle (x − h)2 + (y − k)2 = a2 is x = h + a cos θ , y = k + a sin θ , 0 ≤ θ ≤ 2π
For circle x2 + y2 = a2 , parametric equation is x = a cos θ , y = a sin θ

13. The equation of the tangent at the point P(x1 , y1) to a circle x2 + y2 + 2gx + 2fy + c = 0 is xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
14. The equation of the tangent at the point P(x1 , y1) to a circle x2 + y2 = r2 is xx1 + yy1 = r2.
15. The equation of the tangent of slope m to the circle x2 + y2 + 2gx + 2fy + c = 0 are \tt y+f=m\left(x+g\right)\pm\sqrt{\left(g^{2}+f^{2}-c\right)\left(1+m^{2}\right)}
Parametric Form
1. The equation of the tangent to the circle (x − a)2 + (y − b)2 = r2 at the point (a + r cos θ, b + r sin θ) is (x − a) cos θ + (y − b) sin θ = r.
2. The equation of normal at the point (x1, y1) to the circle x2 + y2 + 2gx + 2fy + c = 0 is
\tt y-y_{1}=\frac{y_{1}+f}{x_{1}+g}\left(x-x_{1}\right)
(y1 + f) x − (x1 + g) y + (gy1 − fx1) = 0
3. The equation of normal at the point (x1 , y1) to the circle x2 + y2 = r2 is \tt \frac{x}{x_{1}}=\frac{y}{y_{1}}.
4. The equation of normal to the circle x 2 + y2 = r2 at the point (r cos θ, r sin θ) is
\tt \frac{x}{r \cos \theta}=\frac{y}{r \sin \theta}
y = x tan θ.