Parametric form

Properties of Focal chord:

1. If the chord joining $\mathrm{P}\left({\mathrm{t}}_1\right)$ and $\mathrm{Q}\left({\mathrm{t}}_2\right)$ is the focal chord then ${\mathrm{t}}_1\cdot {\mathrm{t}}_2=-1$.

$\mathrm{P},\mathrm{S}$ and $\mathrm{Q}$ lies on the focal chord

$\therefore \mathrm{P},\mathrm{S}$ and $\mathrm{Q}$ are collinear slope of PS = slope of SQ

$\begin{array}{rl}& \frac{2{\mathrm{at}}_1}{{\mathrm{at}}_1^2-\mathrm{a}}=\frac{2{\mathrm{at}}_2}{{\mathrm{at}}_2^2-\mathrm{a}}\\ & \frac{2{\mathrm{t}}_1}{{\mathrm{t}}_1^2-1}=\frac{2{\mathrm{t}}_2}{{\mathrm{t}}_2^2-1}\\ & {\mathrm{t}}_1{\mathrm{t}}_2^2-{\mathrm{t}}_1={\mathrm{t}}_2{\mathrm{t}}_1^2-{\mathrm{t}}_2\\ & {\mathrm{t}}_2-{\mathrm{t}}_1={\mathrm{t}}_1{\mathrm{t}}_2({\mathrm{t}}_1-{\mathrm{t}}_2)\\ & -1={\mathrm{t}}_1{\mathrm{t}}_2\text{ or }{\mathrm{t}}_2=-\frac{1}{{\mathrm{t}}_1}\end{array}$

Extremities of a focal chord are $({\mathrm{at}}^2,2\mathrm{at})$ and $\frac{\mathrm{a}}{{\mathrm{t}}^2},\frac{-2\mathrm{a}}{\mathrm{t}}$.

2. Length of focal chord is a $t+{\frac{1}{t}}^2$

$\begin{array}{rl}PQ& =PS+SQ\\ & =a{t}^2+a+\frac{a}{t^2}+a\\ & =a{t}^2+\frac{1}{t^2}+2\\ & =at+{\frac{1}{t}}^2\end{array}$

3. The length of the focal chord which makes an angle $\theta$ with the positive direction of $x$-axis is $4\mathrm{a}{\mathrm{cosec}}^2\theta$.

We know $\mathrm{PQ}=\mathrm{a}\mathrm{t}+{\frac{1}{\mathrm{t}}}^2$

$\begin{array}{rl}& \text{ slope }=\tan \theta =\frac{2\mathrm{at}+\frac{2\mathrm{a}}{\mathrm{t}}}{{\mathrm{at}}^2-\frac{\mathrm{a}}{{\mathrm{t}}^2}}\\ & \text{ slope }=\tan \theta =\frac{2}{\mathrm{t}-\frac{1}{\mathrm{t}}}\\ & 2\cot \theta =\mathrm{t}-\frac{1}{\mathrm{t}}\\ & \therefore \mathrm{PQ}=\mathrm{a}\mathrm{t}+{\frac{1}{\mathrm{t}}}^2\\ & =\mathrm{a}\mathrm{t}-{\frac{1}{\mathrm{t}}}^2+4\\ & =\mathrm{a}[4{\cot }^2\theta +4](\mid \mathrm{cosec}\theta \ge 1)\\ & =4\mathrm{a}{\mathrm{cosec}}^2\theta \end{array}$

• Minimum length of $\mathrm{PQ}=4\mathrm{a}$ (i.e. latus rectum)

4. Semi latus rectum of a parabola is the harmonic mean between the segments of any focal chord of the parabola.

$\begin{array}{rl}& \mathrm{SP}=\mathrm{a}+{\mathrm{at}}^2,\mathrm{SQ}=\mathrm{a}+\frac{\mathrm{a}}{{\mathrm{t}}^2}\\ & \frac{1}{\mathrm{SP}}+\frac{1}{\mathrm{SQ}}=\frac{1}{\mathrm{a}+{\mathrm{at}}^2}+\frac{1}{\mathrm{a}+\frac{\mathrm{a}}{{\mathrm{t}}^2}}=\frac{1}{\mathrm{a}+{\mathrm{at}}^2}+\frac{{\mathrm{t}}^2}{{\mathrm{at}}^2+\mathrm{a}}=\frac{1}{\mathrm{a}}\\ & \therefore 2\mathrm{a}=2x\frac{\mathrm{SP}x\mathrm{SQ}}{\mathrm{SP}+\mathrm{SQ}}\end{array}$

Semi latus rectum $=$ Harmonic Mean of SP and SQ.

5. Circle described on the focal length as diameter touches the tangent at vertex.

Equation of circle PS as diameter is

$(x-a{t}^2)(x-a)+(y-2at)y=0$

Equation of $y$-axis is $x=0$

After solving ${\mathrm{y}}^2-2aty+{\mathrm{a}}^2{\mathrm{t}}^2=0$

$(y-at{)}^2=0$

$\therefore$ circle touches the $y$-axis at $(0$, at $)$.

Example: 1 The length of a focal chord of the parabola $y^2=4ax$ at a distance $b$ from the vertex is $\mathrm{c}$, then

(a). ${\mathrm{b}}^2=4\mathrm{ac}$

(b). $b^2\mathrm{c}={\mathrm{a}}^3$

(c). $b^2\mathrm{c}=4{\mathrm{a}}^3$

(d). $4b^{2}c=a^3$

Show Answer

Solution: $\mathrm{PQ}=4\mathrm{a}{\mathrm{cosec}}^2\theta =\mathrm{c}$

In $\mathrm{△}\mathrm{OMS},\sin \theta =\frac{\mathrm{b}}{\mathrm{a}}$

$\begin{array}{rl}& {\mathrm{cosec}}^2\theta =\frac{{\mathrm{a}}^2}{{\mathrm{b}}^2}\\ \therefore & c=4a\cdot \frac{{\mathrm{a}}^2}{{\mathrm{b}}^2}{b}^{2}c=4{\mathrm{a}}^3\end{array}$

Answer: c

Example: 2 The coordinates of the ends of a focal chord of a parabola $y^2=4ax$ are $(x_1,y_1)$ and $({\mathrm{x}}_2,{\mathrm{y}}_2)$ then value of ${\mathrm{x}}_1{\mathrm{x}}_2+{\mathrm{y}}_1{\mathrm{y}}_2$ is equal to

(a). $3{\mathrm{a}}^2$

(b). $-3a^2$

(c). ${\mathrm{a}}^2$

(d). $-a^2$

Practice questions

1. The focus of the parabola $x^2+8x+12y+4=0$ is

(a). $(4,2)$

(b). $(-2,-4)$

(c). $(2,4)$

(d). $(-4,-2)$

2. The equation of the parabola with vertex at $(3,2)$ and focus at $(5,2)$ is

(a). $x^2-8x-4y-28=0$

(b). $y^2-8x-4y-28=0$

(c). $x^2+8x-4y-28=0$

(d). $y^2+8x+4y-28=0$

3. The equation of the latus rectum of the parabola $x^2+4x+2y=0$ is

(a). $2\mathrm{y}-3=0$

(b). $3\mathrm{y}-2=0$

(c). $2y+3=0$

(d). $3y+2=0$

4. The equation of the parabola whose axis is parallel to $x$-axis and which passes through the points $(0,4),(1,9)$ and $(-2,6)$ is

(a). ${\mathrm{y}}^2+5\mathrm{x}-25\mathrm{y}+139=0$

(b). $3y^2+5x-25y+52=0$

(c). $2{\mathrm{y}}^2-5\mathrm{x}-25\mathrm{y}+68=0$

(d). none of these

5. The parametric equation $x=a^2+bt+c,y=a^{\prime }{t}^2+b^{\prime }t+c^{\prime }$ represents

(a). a circle

(b). a parabola

(c). an ellipse

(d). none of these