Parabola

Tips :

• Equation of a conic with (x₁ y₁) focus lx + my + n = 0 directrix and eccentricity ‘e’ is (l² + m²) [(x - x₁)² + (y - y₁)²] = e² (lx + my + n)²
• A parabola is the locus of a point which moves such that its distance from a fixed point is always equal to its distance from a fixed straight line.
• S = 0 will represent pair of straight lines if \tt \begin{vmatrix}a & h & g \\h & b & f \\ g & f & c\end{vmatrix}=0
• If Δ ≠ 0, h² = ab then s = 0 will represent parabola.
• The equation (y − k)² = 4a (x – h) represents a parabola with axis y − k = 0, vertex (h, k) focus (h + a, k), directrix x = h – a and length of latus rectum 4a.
• If t₁ and t₂ are the extremities of a focal chord of a parabola then t₁ t₂ = -1
• Equation of the tangent to y² = 4ax at (x₁ y₁) is y y₁ = 2a (x + x₁)
• Equation of the tangent y² = 4ax at the point ‘t’ is yt = x + at²
• Condition of Tangency for the parabola y² = 4ax is \tt c= \frac{a}{m}. Here the line is y = mx + c and the point of contact is \tt \left(\frac{a}{m^{2}},\frac{2a}{m}\right)
• Tangent to (y - k)² = ± 4a (x – h) of slope ‘m’ will be the form y - k = m (x – h) ± \tt \frac{a}{m}
• Tangent to (x – h)² = ± 4a (y – k) of slope ‘m’ will be the form y - k = m (x – h) \tt \mp \ am^{2}
• The point of intersection of the tangents at t₁, t₂ on the parabola y² = 4ax is [at₁ t₂, a(t₁ + t₂)]
• The tangents at the ends of a focal chord of the parabola meet on the directrix at right angles.
• The tangent at one end of a focal chord of a parabola is parallel to the normal at the other end.
• The locus of the foot of the perpendicular from the focus to the tangent of a parabola is the tangent at the vertex.
• Equation of common tangent to the parabolas y² = 4ax and x² = 4by is \tt xa^{\frac{1}{3}}+yb^{\frac{1}{3}}+a^{\frac{2}{3}}.b^{\frac{2}{3}}=0
• The maximum number of common normal to y² = 4ax and x² = 4by may be ‘5’ only.
• Area of the triangle formed by the tangents from (x₁ y₁) to the parabola y² = 4ax and its chord of contact is \tt \frac{\left(y_1^2-4a\ x_{1}\right)^{3/2}}{2a}
• The length of chord of contact of tangents drawn from (x₁ y₁) to the parabola y² = 4ax is \tt \frac{\sqrt{\left(y_1^2-4a x_{1}\right)\left(y_1^2+4a^{2}\right)}}{a}
• For any conic y = f(x), length of the chord made by y = mx + c is \tt \sqrt{m^{2}+4c}.\sqrt{1+m^{2}} where ‘m’ is slope of the line.
• From an external point two tangents can be drawn to a parabola. The slopes of the two tangents to y² = 4ax passing through (x₁ y₁) are given by the equation m² x₁ – m y₁ + a = 0
• If m₁, m₂ be the slopes of the two tangents drawn from (x₁ y₁) to y² = 4ax then \tt m_{1}+m_{2}=\frac{y_{1}}{x_{1}},\ m_{1}\ m_{2}=\frac{a}{x_{1}}
• If 'θ' is the angle between the pair of tangents drawn from (x₁ y₁) to y² = 4ax then \tt \tan\theta = \frac{\sqrt{y_1^2-4\ ax_{1}}}{x_{1}+a}\ (or)\ \frac{\sqrt{S_{11}}}{x_{1}+a}
• The locus of point of intersection of the two tangents to y² = 4ax included at a constant angle ‘α’ is (x + a)² tan² α = y² – 4ax
• The condition for lx + my + n = 0 to be a normal to y² = 4ax is al³ + 2al m² + m²n = 0
• Equation of the normal to y² = 4ax having slope ‘m’ is y = mx – 2am – am³ and the foot of the normal is (am², -2am)
• The centroid of the triangle formed by the foot of the ‘3’ normal of the parabola lies on the axis of the parabola.
• If the lines y = mx + c intersect the parabola y² = 4ax in the points A and B then the length of the chord AB is \tt \frac{4}{m^{2}}\sqrt{a\left(a-mc\right)\left(1+m^{2}\right)} and the midpoint of the chord AB is \tt \left(\frac{2a-mc}{m^{2}},\frac{2a}{m}\right)
• If (at², 2at) is one end of a focal chord of the parabola then the other end is \tt \left(\frac{a}{t^{2}},\frac{-2a}{t}\right)
• The length of the focal chord at ‘t’ is \tt a\left(t+\frac{1}{t}\right)^2
• If PSQ is a focal chord of the parabola y² = 4ax with focus ‘S’ then \tt \frac{1}{SP}+\frac{1}{SQ}=\frac{1}{a}
• The least length of a focal chord of a parabola is its length of latusrectum.
• Length of the focal chord of y² = 4ax making an angle θ with its axis is 4a cosec²θ
• Length of the chord of the parabola y² = 4ax passing through the vertex and making an angle ‘θ’ with its axis is 4a cos θ cosec²θ
• The (x₁ y₁) (x₂ y₂) are the extremities of a focal chord of the parabola y² = 4ax then x₁ x₂ = a²; y₁ y₂ = -4a²
• The orthocentre of the triangle formed by the tangents at t₁, t₂, t₃ to the parabola y² = 4ax is (-a, a t₁+t₂+t₃ + t₁ t₂ t₃)
• Area of triangle inscribed in parabola y² = 4ax is \tt \frac{1}{8a}\mid\left(y_{1}-y_{2}\right)\left(y_{2}-y_{3}\right)\left(y_{3}-y_{1}\right)\mid where y₁, y₂ and y₃ are ordinates of angular points.
• Area of triangle formed by tangents drawn to the parabola y² = 4ax at the points whose ordinates are y₁, y₂ and y₃ is \tt \frac{1}{16a}\mid\left(y_{1}-y_{2}\right)\left(y_{2}-y_{3}\right)\left(y_{3}-y_{1}\right)\mid
• The length of the chord joining t₁, t₂ on y² = 4ax is \tt a \mid t_{1}-t_{2}\mid\sqrt{\left(t_{1}+t_{2}\right)^2+4}
• If the normal at ‘t’ on the parabola y² = 4ax subtends a right angle at its focus then t = ± 2 as its vertex then \tt t =\pm\ \sqrt{2}