# 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}

### Latus Rectum in Parabola,Examples

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1. Standard form of the parabola is y2 = 4ax.
2. Parabola opening to left i.e, y2 = −4ax
3. Parabola opening upwards i.e., x2 = 4ay
4. Parabola opening downwards i.e., x2 = −4ay

5.

 Important terms y2 = 4ax y2 = −4ax x2 = 4ay x2 = -4ay Vertex (0, 0) (0, 0) (0, 0) (0, 0) Focus (a, 0) (−a, 0) (0, a) (0, −a) Directrix x = − a x = a y = − a y = a Axis y = 0 y = 0 x = 0 x = 0 Latustrectum 4a 4a 4a 4a Focal distance P(x, y) x + a x − a y + a a − y

6. If the vertex of the parabola is at a point A(h, k) and its latustrectum is of length 4a, then its equation is
a) (y − k)2 = 4a (x − h), its axis is parallel to OX i.e., parabola open rightward.
b) (y − k)2 = − 4a (x − h), its axis is parallel to OX' i.e., parabola open leftward.
c) (x − h)2 = − 4a (y − k), its axis is parallel to OY i.e., parabola open upward.
d) (x − h)2 = − 4a (y − k), its axis is parallel to OY' i.e., parabola open downward.
7. Point Form
The equation of the tangent to the parabola y2 = 4ax at a point (x1 , y1) is given by yy1= 2a (x + x1).

8. Slope Form
a) The equation of the tangent of slope m to the parabola y2 = 4ax is \tt y=mx+\frac{a}{m}
b) The equation of the tangent of slope m to the parabola (y − k)2 = 4a (x − h) is given by (y − k) = m (x − h) + \tt \frac{a}{m}
c) The coordinates of the point of contact are \tt \left(h+\frac{a}{m^{2}},k+\frac{2a}{m}\right)
9. Parametric Form: The equation of the tangent to the parabola y2 = 4ax at a point (at2 , 2at) is yt = x + at2.
10. Point Form: The equation of the normal to the parabola y2 = 4ax at a point (x1 , y1) is given by y − y1 = \tt -\frac{y_{1}}{2a}\left(x-x_{1}\right).
11. Parametric Form : The equation of the normal to the parabola y2 = 4ax at point (at2 , 2at) is given by y + tx = 2at + at3.
12. Slope Form: The equation of the normal to the parabola y2 = 4ax in terms of its slope m is given by y = mx − 2am − am3 at point (am2 , −2am).