## Conic Sections

# 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.
- ax
^{2}+ 2hxy + by^{2}+ 2gx + 2fy + c = 0 represents a circle if h=0; a = b ≠ 0 and g^{2}+ f^{2}– ac ≥ 0 - General equation of standard form of a circle is x
^{2}+ y^{2}+ 2gx + 2fy + c = 0 if g^{2}+ f^{2}– c > 0 then real circle, if g^{2}+ f^{2}– c = 0 then point circle, if g^{2}+ f^{2}– c < 0 then imaginary circle. - The general equation of a circle is ax
^{2}+ ay^{2}+ 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(x
^{2}+ y^{2}) + c(bx + ay) = 0 - The equation of the circle with (x
_{1}y_{1}) and (x_{2}y_{2}) a extremities of a diameter is (x - x_{1}) (x - x_{2})+ (y – y_{1}) (y – y_{2}) = 0 - Any circle touching x-axis is of the form x
^{2}+ y^{2}+2gx + 2fy + g^{2}= 0. [c = g^{2}] radius = |f|; if it is touching both axes then c = g^{2}= f^{2}and radius = |g| = |f|. - Conditions for a line is tangent to a circle y = mx + c touches x
^{2}+ y^{2}= r^{2}is c^{2}= r^{2}(1 + m^{2}) - Suppose lx + my + n = 0 touches x
^{2}+ y^{2}= r^{2}then n^{2}= r^{2}(l^{2}+ m^{2}) and the point of contact is \tt \left(\frac{-lr^{2}}{n},\frac{-mr^{2}}{n}\right) - Suppose lx + my + n = 0 touches x
^{2}+ y^{2}+2gx + 2fy + c = 0 is (l^{2}+ m^{2}) (g^{2}+ f^{2}– c) = (lg + mf – n)^{2} - The equation of tangent to the circle x
^{2}+ y^{2}= r^{2}whose slope ‘m’ is \tt y = mx\ \pm\ r\sqrt{m^{2}+1} - The intercept made by the circle x
^{2}+ y^{2}+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 x
^{2}+ y^{2}= r^{2}is x cos θ + y sin θ = r - Equation of the normal at ‘θ’ to the circle x
^{2}+ y^{2}= r^{2}is x sin θ – y cos θ = 0 - The equation of the normal at P(x
_{1}y_{1}) to x^{2}+ y^{2}+2gx + 2fy + c = 0 is y_{1}(x + g) – x_{1}(y + f) = 0 - Equation of the chord joining the two points θ
_{1}and θ_{2}of the circle x^{2}+ y^{2}= r^{2}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(θ
_{1}), B(θ_{2}) of the circle x^{2}+ y^{2}= r^{2}(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 CP
^{2}– r^{2} - 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(x
_{1}y_{1}) 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 x
^{2}+ y^{2}= r^{2}is \left(\frac{-lr^{2}}{n},\frac{-mr^{2}}{n}\right) - The condition that the lines l
_{1}x + m_{1}y + n_{1}= 0 and l_{2}x + m_{2}y + n_{2}= 0 to be conjugate w.r.t the circle x^{2}+ y^{2}= r^{2}is r^{2}(l_{1}l_{2}+ m_{1}m_{2}) = n_{1}n_{2} - The condition that the lines l
_{1}x + m_{1}y + n_{1}= 0 and l_{2}x + m_{2}y + n_{2}= 0 to be conjugate w.r.t the circle x^{2}+ y^{2}+ 2gx + 2fy + c = 0 is (g^{2}+ f^{2}– c) (l_{1}l_{2}+ m_{1}m_{2}) = (l_{1}g + m_{1}f – n_{1}) (l_{2}g + m_{2}f – n_{2}) - If 'c' is centre 'r' is radius of a circle and P, Q are a pair of inverse points then CP.CQ = r
^{2} - The inverse point of (x
_{1}y_{1}) with respect to circle x^{2}+ y^{2}= r^{2}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 x
^{2}+ y^{2}+ 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 x
^{2}+ y^{2}+ 2gx + 2fy + c = 0 is (gx + fy)^{2}= c (x^{2}+ y^{2}) - 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(x
_{1}y_{1}) 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(x
_{1}y_{1}) 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 x
^{2}+ y^{2}= r^{2}is x^{2}+ y^{2}= r^{2}cosec^{2}θ/2 - The locus of midpoints of chords of circle x
^{2}+ y^{2}= r^{2}which subtend angle ‘θ’ at centre is x^{2}+ y^{2}= r^{2}cos^{2}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 x
^{2}+ y^{2}= a^{2}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 x
^{2}+ y^{2}+ 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 x
^{2}+ y^{2}+ ax + by + c = 0 and x^{2}+ y^{2}+ 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 x
^{2}+ y^{2}+ 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 x
^{2}+ y^{2}= r^{2}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

### Introduction to Conic Sections

### Introduction to Circle

### Circle Examples

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1. The general equation of a circle is given by x^{2} + y^{2} + 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} = a^{2}.

If centre is (0, 0), then equation of circle is x^{2} + y^{2} = a^{2}.

3. When the circle passes through the origin, then equation of the circle is x^{2} + y^{2} − 2hx − 2ky = 0.

4. When the circle touches the X-axis, the equation is x^{2} + y^{2} − 2hx − 2ay + h^{2} = 0.

5. Equation of the circle, touching the Y-axis is x^{2} + y^{2} − 2ax − 2ky + k^{2} = 0.

6. Equation of the circle, touching both axes is x^{2} + y^{2} − 2ax − 2ay + a^{2} = 0.

7. Equation of the circle passing through the origin and centre lying on the X-axis is x^{2} + y^{2} − 2ax = 0.

8. Equation of the circle passing through the origin and centre lying on the Y-axis is x^{2} + y^{2} − 2ay = 0.

9. Equation of the circle through the origin and cutting intercepts a and b on the coordinate axes is x^{2} + y^{2} − by = 0.

10. Equation of the circle, when the coordinates of end points of a diameter are (x_{1} , y_{1}) and (x_{2} , y_{2}) is

(x − x_{1}) (x − x_{2}) + (y − y_{1}) (y − y_{2}) = 0.

11. Equation of the circle passes through three given points (x_{1} , y_{1}) , (x_{2} , y_{2}) and (x_{3} , y_{3}) 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} = a^{2} is x = h + a cos θ , y = k + a sin θ , 0 ≤ θ ≤ 2π

For circle x^{2} + y^{2} = a^{2} , parametric equation is x = a cos θ , y = a sin θ

13. The equation of the tangent at the point P(x_{1} , y_{1}) to a circle x^{2 }+ y^{2} + 2gx + 2fy + c = 0 is xx_{1} + yy_{1} + g(x + x_{1}) + f(y + y_{1}) + c = 0

14. The equation of the tangent at the point P(x_{1} , y_{1}) to a circle x^{2} + y^{2} = r^{2} is xx_{1} + yy_{1} = r^{2}.

15. The equation of the tangent of slope m to the circle x^{2} + y^{2} + 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} = r^{2} 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 (x_{1}, y_{1}) to the circle x^{2} + y^{2} + 2gx + 2fy + c = 0 is

\tt y-y_{1}=\frac{y_{1}+f}{x_{1}+g}\left(x-x_{1}\right)

(y_{1} + f) x − (x_{1} + g) y + (gy_{1} − fx_{1}) = 0

3. The equation of normal at the point (x_{1} , y_{1}) to the circle x^{2} + y^{2} = r^{2} is \tt \frac{x}{x_{1}}=\frac{y}{y_{1}}.

4. The equation of normal to the circle x ^{2} + y^{2} = r^{2} 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.