Equations of Normals in different forms

(i) Point form

Let equation of ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Equation of normal is $\frac{a^{2}x}{x_1}-\frac{b^{2}y}{y_1}=a^2-b^2$

(ii) Parametric form

Let equation of ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Equation of normal is $ax\sec \varphi -by\text{cosec}\varphi ={\mathrm{a}}^2-{\mathrm{b}}^2$

$(a\cos \varphi ,b\sin \varphi )$ is parametric coordinates of $\mathrm{P}({\mathrm{x}}_1,{\mathrm{y}}_1)$, i.e. $\mathrm{P}(\mathrm{a}\cos \varphi ,\mathrm{b}\sin \varphi )$

(iii) Slope form

For an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Equation of normal in terms of slope $(\mathrm{m})$ is

$y=mx\pm \frac{m(a^2-b^2)}{\sqrt{a^2+b^2{m}^2}}$

Condition of normality when $y=mx+c$ is the normal of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

$\begin{array}{r}{\mathrm{c}}^2=\frac{{\mathrm{m}}^2{({\mathrm{a}}^2-{\mathrm{b}}^2)}^2}{{\mathrm{a}}^2+{\mathrm{b}}^2{\mathrm{m}}^2}\\ \text{ or }\mathrm{c}=\pm \frac{\mathrm{m}({\mathrm{a}}^2-{\mathrm{b}}^2)}{\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2{\mathrm{m}}^2}}\end{array}$

Examples

1. If the normal at the point $P(\theta )$ to the ellipse $\frac{x^2}{14}+\frac{y}{5}=1$ intersects it again at the point $Q(2\theta )$ then $\cos \theta$ is equal to

(a) $\frac{2}{3}$

(b) $-\frac{2}{3}$

(c) $\frac{3}{2}$

(d) $-\frac{3}{2}$

2. The equation of the normal to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the end of the latus rectum in the first quadrant is

(a) $x+$ ey $-a{e}^3=0$

(b) $\mathrm{x}-\mathrm{ey}+\mathrm{ae}{\mathrm{e}}^3=0$

(c) $\mathrm{x}-\mathrm{ey}-{\mathrm{ae}}^3=0$

(d) None of these

3. The condition that the line $x\cos \alpha +y\sin \alpha =p$ may be a normal to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ then

(a) ${(a^2-b^2)}^2=p^2(a^2{\sec }^2\alpha +b^2{\mathrm{cosec}}^2\alpha )$

(c) ${(a^2+b^2)}^2=p^2(a^2{\sec }^2\alpha +b^2{\mathrm{cosec}}^2\alpha )$

(b) ${({\mathrm{a}}^2-{\mathrm{b}}^2)}^2={\mathrm{p}}^2({\mathrm{a}}^2{\mathrm{cosec}}^2\alpha +{\mathrm{b}}^2{\sec }^2\alpha )$

(d) None of these

4. If the normals at $\mathrm{P}({\mathrm{x}}_1,{\mathrm{y}}_1),\mathrm{Q}({\mathrm{x}}_2,{\mathrm{y}}_2)$ and $\mathrm{R}({\mathrm{x}}_3,{\mathrm{y}}_3)$ to the ellipse are concurrent, then

(a) $\left|\begin{array}{lll}{x}_2& y_2& x_2{y}_2\\ x_1& y_1& x_1{y}_1\\ x_3& y_3& x_3{y}_3\end{array}\right|=-1$

(b) $\left|\begin{array}{lll}{\mathrm{x}}_1& {\mathrm{y}}_1& {\mathrm{x}}_1{\mathrm{y}}_1\\ {\mathrm{x}}_2& {\mathrm{y}}_2& {\mathrm{x}}_2{\mathrm{y}}_2\\ {\mathrm{x}}_3& {\mathrm{y}}_3& {\mathrm{x}}_3{\mathrm{y}}_3\end{array}\right|=0$

(c) $\left|\begin{array}{lll}{x}_2& y_2& x_2{y}_2\\ x_3& y_3& x_3{y}_3\\ x_1& y_1& x_1{y}_1\end{array}\right|=1$

(d) None of these

Practice questions

1. In the normal at the end of latus rectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity e, passes through one end of the minor axis, then :

(a) ${\mathrm{e}}^2(1+{\mathrm{e}}^2)=0$

(b) ${\mathrm{e}}^2(1+{\mathrm{e}}^2)=1$

(c) ${\mathrm{e}}^2(1+{\mathrm{e}}^2)=-1$

(d) ${\mathrm{e}}^2(1+{\mathrm{e}}^2)=2$

2. If the normals to $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ at the ends of the chords ${\ell }_1\mathrm{x}+{\mathrm{m}}_1\mathrm{y}=1$ and ${\ell }_2\mathrm{x}+{\mathrm{m}}_2\mathrm{y}=1$ are concurrent, then :

(a) ${\mathrm{a}}^2{\ell }_1{\ell }_2+{\mathrm{b}}^2{\mathrm{m}}_1{\mathrm{m}}_2=1$

(b) ${\mathrm{a}}^2{\ell }_1{\ell }_2+{\mathrm{b}}_2{\mathrm{m}}_1{\mathrm{m}}_2=-1$

(c) ${\mathrm{a}}^2{\ell }_1{\ell }_2-{\mathrm{b}}^2{\mathrm{m}}_1{\mathrm{m}}_2=-1$

(d) None of these

3. If the normal at an end of a latus rectum of an ellipse passes through one extremity of the minor axis, then the eccentricity of the ellipse is given by

(a) ${\mathrm{e}}^4+{\mathrm{e}}^2-1=0$

(b) ${\mathrm{e}}^4+{\mathrm{e}}^2-5=0$

(c) ${\mathrm{e}}^3=\sqrt{5}$

(d) None of these

4. The number of normals that can be drawn from a point to a given ellipse is

(a) 2

(b) 3

(c) 4

(d) 1

5. If the normal at any point $\mathrm{P}$ on the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ meets the axes in $\mathrm{G}$ and $\mathrm{g}$ respectively, then $\mathrm{PG}:\mathrm{Pg}$ is equal to

(a) $a:b$

(b) $a^2:b^2$

(c) $b^2:a^2$

(d) $\mathrm{b}:\mathrm{a}$

6. If normal to ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at $(ae,\frac{b^2}{a})$ is passing through $(0,-2b)$, then value of eccentricity is

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

(b) $2(\sqrt{2}-1)$

(c) $\sqrt{2(\sqrt{2}-1)}$

(d) None of these

7. If normal at any point $\mathrm{P}$ to the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1,\mathrm{a}>\mathrm{b}$ meet the axes at $\mathrm{M}$ and $\mathrm{N}$ so that $\frac{\mathrm{PM}}{\mathrm{PN}}=\frac{2}{3}$, then value of eccentricity e is

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

(b) $\sqrt{\frac{2}{3}}$

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

(d) None of these

8. If the tangent drown at point $(t^2,2t)$ on the parabola $y^2=4\mathrm{x}$ is same as the normal drawn at point $(\sqrt{5}\cos \theta ,2\sin \theta )$ on the ellipse $4x^2+5y^2=20$. Then the values of $\mathrm{tand}\theta$ are

(a) $\theta ={\cos }^{-1}\left(\frac{-1}{\sqrt{5}}\right)\mathrm{\&}\mathrm{t}=\frac{-1}{\sqrt{5}}$

(b) $\theta ={\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)\mathrm{\&}\mathrm{t}=\frac{1}{\sqrt{5}}$

(c) $\theta ={\cos }^{-1}\left(\frac{-2}{\sqrt{5}}\right)\mathrm{\&}\mathrm{t}=\frac{-2}{\sqrt{5}}$

(d) None of these

9. The normals at four points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meet in the point $(h,k)$. Then the mean position of the four points is

(a) $(\frac{a^{2}h}{2(a^2+b^2)},\frac{b^{2}k}{2(a^2+b^2)})$

(b) $(\frac{a^{3}h}{2(a^2+b^2)},\frac{b^{3}k}{2(a^2+b^2)})$

(c) $(\frac{\mathrm{ah}}{2({\mathrm{a}}^2-{\mathrm{b}}^2)},\frac{\mathrm{bk}}{2({\mathrm{a}}^2-{\mathrm{b}}^2)})$

(d) $(\frac{a^{2}h}{2(a^2-b^2)},\frac{b^{2}k}{2(a^2-b^2)})$

10. The equation of the normal at the point $(2,3)$ on the ellipse $9x^2+16y^2=180$ is

(a) $3y=8x-10$

(b) $3y-8x+7=0$

(c) $8y+3x+7=0$

(d) $3x+2y+7=0$

11. Number of distinct normal lines that can be drawn to the ellipse $\frac{x^2}{169}+\frac{y^2}{25}=1$ from the point $P$ $(0,6)$ is

(a) one

(b) two

(c) three

(d) four

12. Any ordinate MP of the ellipse $\frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{9}=1$ meets the auxiliary circle at $\mathrm{Q}$, then locus of the point of intersection of normals at $\mathrm{P}$ and $\mathrm{Q}$ to the respective curve is

(a) ${\mathrm{x}}^2+{\mathrm{y}}^2=8$

(b) $x^2+y^2=34$

(c) ${\mathrm{x}}^2+{\mathrm{y}}^2=64$

(d) $x^2+y^2=15$

13. If the normals at $\mathrm{P}(\theta )$ and $\mathrm{Q}(\frac{\pi }{2}+\theta )$ to the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ meet the major axis at $\mathrm{G}$ and $\mathrm{g}$ respectively, then ${\mathrm{PG}}^2+{\mathrm{Qg}}^2=$

(a) ${\mathrm{b}}^2(1-{\mathrm{e}}^2)(2-{\mathrm{e}}^2)$

(b) ${\mathrm{a}}^2({\mathrm{e}}^4-{\mathrm{e}}^2+2)$

(c) ${\mathrm{a}}^2(1+{\mathrm{e}}^2)(2+{\mathrm{e}}^2)$

(d) ${\mathrm{b}}^2(1+{\mathrm{e}}^2)(2+{\mathrm{e}}^2)$