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