Equation of Normal

Equation of normal to the parabola $(y-k{)}^2=4(x-h)$ is

$y-k=m(x-h)-2am-a^3$

Properties of Normal

1. If the normal at the point $\mathrm{P}({\mathrm{at}}_1{}^2,2{\mathrm{at}}_1)$ meets the parabola at

$\mathrm{Q}({\mathrm{at}}_2{}^2,2{\mathrm{at}}_2)$, then ${\mathrm{t}}_2=-{\mathrm{t}}_1-{\displaystyle \frac{2}{{\mathrm{t}}_1}}$

2. If the normal at the points $(a{t}_1{}^2,2{\mathrm{at}}_1)$ and $({\mathrm{at}}_2{}^2,2{\mathrm{at}}_2)$ meet on the parabola ${\mathrm{y}}^2=4\mathrm{ax}$, then ${\mathrm{t}}_1{\mathrm{t}}_2=2$.

3. No normal other than axis passes through focus.

Important Properties :

• If the tangent and normal at any point ’ $\mathrm{P}$ ’ of the parabola intersect the axis at $\mathrm{T}$ and $\mathrm{N}$ then $\mathrm{ST}$ $=\mathrm{SN}=\mathrm{SP}$ where $\mathrm{S}$ in the focus.

• The portion of a tangent to a parabola cut off between the directrix & the curve subtends a right angle at the focus.

$\mathrm{\angle }\mathrm{PSQ}={90}^{\circ }$

• Any tangent to a parabola and the perpendicular on it from the focus meet on the tangent at the vertex.

$\mathrm{\angle }\mathrm{PQS}={90}^{\circ }$

• If the tangents at $\mathrm{A}$ and $\mathrm{B}$ meet in $\mathrm{P}$ then $\mathrm{PA}$ and $\mathrm{PB}$ subtends equal angles at the focus $\mathrm{S}$. $(\mathrm{SP}{)}^2=\mathrm{SA}\times \mathrm{SB}$

$\mathrm{△}\mathrm{SAP}\sim \mathrm{△}\mathrm{SPB}$

$\mathrm{\angle }\mathrm{PSA}=\mathrm{\angle }\mathrm{PSB}$.

• The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.

Conormal points:

Let $\mathrm{P}(\mathrm{h},\mathrm{k})$ be a point and equation of parabola be ${\mathrm{y}}^2=4\mathrm{ax}$.

Equation of normal is

$y=mx-2am-a{m}^3$

If passes through $(\mathrm{h},\mathrm{k})$ so

$\begin{array}{rl}& \mathrm{k}=\mathrm{mh}-2\mathrm{am}-{\mathrm{am}}^3\\ & {\mathrm{am}}^3+2\mathrm{am}-\mathrm{mh}+\mathrm{k}=0\end{array}$

${\mathrm{am}}^3+\mathrm{m}(2\mathrm{a}-\mathrm{h})+\mathrm{k}=0$

Suppose ${\mathrm{m}}_1,{\mathrm{m}}_2,{\mathrm{m}}_3$ are the roots of this equation

$\therefore {\mathrm{m}}_1+{\mathrm{m}}_2+{\mathrm{m}}_3=0$

$\begin{array}{rl}& {\mathrm{m}}_1{\mathrm{m}}_2+{\mathrm{m}}_2{\mathrm{m}}_3+{\mathrm{m}}_3{\mathrm{m}}_1={\displaystyle \frac{2\mathrm{a}-\mathrm{h}}{\mathrm{a}}}\\ & {\mathrm{m}}_1\cdot {\mathrm{m}}_2\cdot {\mathrm{m}}_3=-{\displaystyle \frac{\mathrm{k}}{\mathrm{a}}}\end{array}$

So maximum three normal say PM, PN, PQ drawn through P. Points M, N, Q are called conormal points.

• The algebraic sum of ordinates of the conormal points is zero.

Let the coordinates of conormal points be $\mathrm{M}({\mathrm{am}}_1{}^2,-2{\mathrm{am}}_1),\mathrm{N}({\mathrm{am}}_2{}^2,-2{\mathrm{am}}_2)$ and

$\mathrm{Q}({\mathrm{am}}_3^2,-2{\mathrm{am}}_3)$. The ordinates of these points

${\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3=-2{\mathrm{am}}_1-2{\mathrm{am}}_2-2{\mathrm{am}}_3$

$=-2\mathrm{a}({\mathrm{m}}_1+{\mathrm{m}}_2+{\mathrm{m}}_3)$

$=0$

$\Rightarrow {\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3=0$

• Centroid of the triangle formed by conormal points lies on the axis of parabola.

Let coordinates of conormal points be $\mathrm{M}({\mathrm{x}}_2,{\mathrm{y}}_1),\mathrm{N}({\mathrm{x}}_2,{\mathrm{y}}_2)\mathrm{Q}({\mathrm{x}}_3,{\mathrm{y}}_3)$

Then centroid is $({\displaystyle \frac{{\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3}{3}},{\displaystyle \frac{{\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3}{3}})=({\displaystyle \frac{{\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3}{3}},0)$

Since sum of ordinates is zero. Therefore centroid lies on the axis of parabola.

• $$ Normal drown from a point $\mathrm{P}(\mathrm{h},\mathrm{k})$ to the parabola are real and distinct if $\mathrm{h}>2\mathrm{a}$.

Chord of Contact

Let $\mathrm{PA}$ and $\mathrm{PB}$ be tangents drawn through the point $\mathrm{P}(\mathrm{h},\mathrm{k})$.

Equation of tangent at $\mathrm{A}$ is

${\mathrm{yy}}_1=2\mathrm{a}(\mathrm{x}+{\mathrm{x}}_1)$

Equation of chord whose midpoint $(x_1,y_1)$ is given :

$\begin{array}{rl}& \mathrm{S}\equiv {\mathrm{y}}^2-4\mathrm{ax}\\ & {\mathrm{S}}_1\equiv {\mathrm{y}}_1{}^2-4\mathrm{ax}\\ & \mathrm{T}\equiv {\mathrm{y}}_1-2\mathrm{a}(\mathrm{x}+{\mathrm{x}}_1)\end{array}$

Equation of $\mathrm{AB}$ is $\mathrm{T}=\mathrm{S}$

PI is incident ray then PS is reflected ray. So any ray incident parallel to axis of the parabola after reflection it passes through focus.

Example: 17 If the chord of contact of tangent from a point $\mathrm{P}$ to the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ touches the parabola ${\mathrm{x}}^2=4\mathrm{by}$. The locus of $\mathrm{P}$ is

(a) Parabola

(b) Hyperbola

(c) ellipse

(d) Circle

Example: 18 Let $P$ and $Q$ be points $(4,-4)$ and $(9,6)$ on the parabola $y^2=4a(x-b)$. $R$ is a point on the parabola so that area $\mathrm{△}\mathrm{PRQ}$ is maximum, then

(a) $\mathrm{\angle }\mathrm{PRQ}={90}^{\circ }$

(b) the point $\mathrm{R}$ is $(4,4)$

(c) the point $\mathrm{R}$ is $({\displaystyle \frac{1}{4}},1)$

(d) None of these

Example: 19 Minimum area of circle which touches the parabola’s $y^2=x^2+1$ and $y^2=x+1$ is

(a) ${\displaystyle \frac{9\pi }{32}}$ sq.unit

(b) ${\displaystyle \frac{\pi }{4}}$ sq.unit

(c) ${\displaystyle \frac{7\pi }{32}}$ sq.unit

(d) ${\displaystyle \frac{9\pi }{16}}$ sq.unit

Show Answer

Solution: $\mathrm{y}={\mathrm{x}}^2+1$ and ${\mathrm{y}}^2=\mathrm{x}+1$ are symmetrical about $\mathrm{y}=\mathrm{x}$

tangent at point. $\mathrm{A}$ and $\mathrm{B}$ are parallel to the line $\mathrm{y}=\mathrm{x}$

$\mathrm{y}={\mathrm{x}}^2+1$ ${\mathrm{y}}^2=\mathrm{x}+1$

${\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}=2\mathrm{x}=1$ $\mathrm{y}={\displaystyle \frac{1}{2}}$

$\mathrm{x}={\displaystyle \frac{1}{2}}$ $\mathrm{x}={\displaystyle \frac{5}{4}}$

$\mathrm{y}={\displaystyle \frac{5}{4}}$

$\mathrm{A}({\displaystyle \frac{1}{2}},{\displaystyle \frac{5}{4}})\mathrm{B}({\displaystyle \frac{5}{4}},{\displaystyle \frac{1}{2}})$

$\mathrm{AB}=\sqrt{{({\displaystyle \frac{1}{2}}-{\displaystyle \frac{5}{4}})}^2+{({\displaystyle \frac{5}{4}}-{\displaystyle \frac{1}{2}})}^2}={\displaystyle \frac{3\sqrt{2}}{4}}$

Area of circle $=\pi {\mathrm{r}}^2=\pi {\left({\displaystyle \frac{3\sqrt{2}}{8}}\right)}^2={\displaystyle \frac{9\pi }{32}}$ sq.unit

Answer: a

Example: 20 The equation of the common tangents to the parabola $\mathrm{y}={\mathrm{x}}^2$ and $\mathrm{y}=-(\mathrm{x}-2{)}^2$ is $/$ are

(a) $\mathrm{y}=4(\mathrm{x}-1)$

(b) $y=0$

(c) $y=-4(x-1)$

(d) $y=-30x-50$

Example 21. $(3,0)$ is the point from which three normals are drawn to the parabola $y^2=4x$ which meet the parabola in the points $\mathrm{P},\mathrm{Q}$ and $\mathrm{R}$ then

Comprehension based Questions (Exampels 6 to 8)

Comprehension 1

Consider the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=9$ and the parabola ${\mathrm{y}}^2=8\mathrm{x}$. They intersect at $\mathrm{P}$ and $\mathrm{Q}$ in the first and the fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $x$-axis at $R$ and tangents to the parabola at $\mathrm{P}$ and $\mathrm{Q}$ intersect the $\mathrm{x}$-axis at $\mathrm{S}$.

Example 22. The ratio of the area of the triangles $\mathrm{PQS}$ and $\mathrm{PQR}$ is

(a) $1:\sqrt{2}$

(b) 1:2

(c) $1:4$

(d) $1:8$

Example 23. The radius of the circum circle of the triangle PRS is

(a) 5

(b) $3\sqrt{3}$

(c) $3\sqrt{2}$

(d) $2\sqrt{3}$

Example 24. The radius of the in circle of the triangle $\mathrm{PQR}$ is

(a) 4

(b) 3

(c) ${\displaystyle \frac{8}{3}}$

(d) 2

COMPREHENSION 2 (EXAMPLES 25 TO 27)

If $y=x$ is tangent to the parabola $y=a{x}^2+c$

25. If $a=2$, then the value of $c$ is

(a) ${\displaystyle \frac{1}{2}}$

(b) ${\displaystyle \frac{1}{4}}$

(c) ${\displaystyle \frac{1}{8}}$

(d) 1

26. If $(1,1)$ is point of contact then ’ $a$ ’ is

(a) 1

(b) ${\displaystyle \frac{1}{2}}$

(c) ${\displaystyle \frac{1}{3}}$

(d) ${\displaystyle \frac{1}{4}}$

27. If $c=2$ then point of contact is

(a) $(4,4)$

(b) $(2,2)$

(c) $(8,8)$

(d) $({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}})$

Exercise

1. The point $\mathrm{P}$ on the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ for which $|\mathrm{PR}-\mathrm{PQ}|$ is maximum, where $\mathrm{R}(-\mathrm{a},0),\mathrm{Q}(0,\mathrm{a})$ is

(a) $(4a,-4a)$

(b) $(4a,4a)$

(c) $(a,2a)$

(d) $(\mathrm{a},-2\mathrm{a})$

2. The shortest distance between the parabola $y^2=4x$ and the circle $x^2+y^2+6x-12y+20=0$ is

(a) $4\sqrt{2}-5$

(b) $4\sqrt{2}+5$

(c) $3\sqrt{2}+5$

(d) $3\sqrt{2}-5$

3. If normals are drawn from a point $p(h,k)$ to the parabola $y^2=4ax$, then the sum of the intercepts which the normals act off from the axis of the parabola is

(a) $4(\mathrm{h}+0)$

(b) $3(\mathrm{h}+\mathrm{c})$

(c) $2(\mathrm{h}+\mathrm{a})$

(d) $(\mathrm{h}+\mathrm{a})$

4. If $a\ne 0$ and the line $2px+3qy+4r=0$ passes through the points of intersection of the parabolas $y^2=4ax$ and $x^2=4ay$, then

(a) ${\mathrm{r}}^2+(3\mathrm{p}+2\mathrm{q}{)}^2=0$

(b) $r^2+(2p+3q{)}^2=0$

(c) ${\mathrm{r}}^2+(3\mathrm{p}-2\mathrm{q}{)}^2=0$

(d) $r^2+(2p-2q{)}^2=0$

5. The equation of the tangent at the vertex of the parabola $x^2+4x+2y=0$ is

(a) $\mathrm{x}=2$

(b) $x=2$

(c) $y=2$

(d) $y=2$

6. The common tangent to the parabolas $y^2=4ax$ and $x^2=4ax$ and $x^2=32ay$ is

(a) $x+2y-4a=0$

(b) $x+2y+4a=0$

(c) $x-2y+4a=0$

(d) $x-2y-4a=0$

7. The shortest distnae between the parabolas ${\mathrm{y}}^2=4\mathrm{x}$ and ${\mathrm{y}}^2=2\mathrm{x}-6$ is

(a) $\sqrt{5}$

(b) 2

(c) 3

(d) none of these

8. The largest value of a for which the circle $x^2+y^2=a^2$ falls totally in the interior of the parabola $y^2=4(x+4)$ is

(a) 4

(b) $4\sqrt{3}$

(c) $3\sqrt{3}$

(d) $2\sqrt{3}$

Multiple choice questions with one or more than one correct answer.

9. Let $P(x_1,y_1)$ and $Q(x_2,y_2),y_1<0,y_2,0$, be the end points of the latus rectum of the ellipse $x^2+4y=4$. The equations of parabolas with latus rectum $\mathrm{PQ}$ are

(a) $x^2+2\sqrt{3}\mathrm{y}=3+\sqrt{3}$

(b) $x^2-2\sqrt{3}y=3+\sqrt{3}$

(c) $x^2+2\sqrt{3}y=3-\sqrt{3}$

(d) $x^2-2\sqrt{3}y=3-\sqrt{3}$

10. The tangent $\mathrm{PT}$ and the normal $\mathrm{PN}$ to the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ at a point $\mathrm{P}$ on it meet its axis at point $\mathrm{T}$ and N, respectively. The locus of the centroid of the triangle PTN is a parabola whose

(a) vertex is $({\displaystyle \frac{2\mathrm{a}}{3}},0)$

(b) directrix is $x=0$

(c) latus rectum is ${\displaystyle \frac{2\mathrm{a}}{3}}$

(d) focus is $(a,0)$

11. Match the following :

Consider the parabola ${\mathrm{y}}^2=12\mathrm{x}$

Assertion and Reasoning

12. Statement 1 : The curve $y={\displaystyle \frac{x^2}{2}}+x+1$ is symmetric with respect to the line $x=1$.

Statement 2 : A parabola is symmetric about its axis.

(A) Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanations for statement 1 .

(B) Statement 1 is True, statement 2 is true, statement 2 is not a correct explanation for statement 1 .

(C) Statement 1 is true, statement 2 is false.

(D) Statement 1 is false, statement 2 is true.