Intersection of a line and an ellipse

Line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ $ …….(1) \text {and ellipse} \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1……(2)$

Solving equations (1) & (2) we get

$\left(a^{2} m^{2}+b^{2}\right) x^{2}+2 a^{2} \mid c m x+a^{2}\left(c^{2}-b^{2}\right)=0$

If $D>0$ then $y=m x+c$ is a secant $\mathrm{D}=0$ then $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ is a tangent $\mathrm{D}<0 \mathrm{y}=\mathrm{mx}+\mathrm{c}$ does not meet ellipse

Point form

Equation of tangent to the ellipse at point $\left(x _{1}, y _{1}\right)$

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

Then equation of tangent in point form is $\frac{\mathrm{xx} _{1}}{\mathrm{a}^{2}}+\frac{\mathrm{yy} _{1}}{\mathrm{~b}^{2}}=1$

Parametric form

Equation of tangent at point $(a \cos \theta, b \sin \theta)$ to the ellipse is $\frac{x \cos \theta}{a}+\frac{y \sin \theta}{b}=1$

Slope form

$y=m x \pm \sqrt{a^{2} m^{2}+b^{2}}$ is a tangent to an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ and point of contact is $\left( \pm \frac{a^{2} m}{\sqrt{a^{2} m^{2}+b^{2}}}, \overline{+} \frac{b^{2}}{\sqrt{a^{2} m^{2}+b^{2}}}\right)$

Number of tangents through a given point $P(h, k)$

$y=m x+\sqrt{a^{2} m^{2}+b^{2}}$ is any tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

If it passes through $\mathrm{P}(\mathrm{h}, \mathrm{k})$ then

$\mathrm{k}=\mathrm{mh}+\sqrt{\mathrm{a}^{2} \mathrm{~m}^{2}+\mathrm{b}^{2}}$

$\mathrm{k}-\mathrm{mh}=\sqrt{\mathrm{a}^{2} \mathrm{~m}^{2}+\mathrm{b}^{2}}$

$(\mathrm{k}-\mathrm{mh})^{2}=\mathrm{a}^{2} \mathrm{~m}^{2}+\mathrm{b}^{2}$

$\mathrm{m}^{2}\left(\mathrm{~h}^{2}-\mathrm{a}^{2}\right)-2 \mathrm{hkm}+\left(\mathrm{k}^{2}-\mathrm{b}^{2}\right)=0$

It is a quadratic in $m$ and will give two values of $m$ hence there are two tangents.

Examples

1. If the line $3 x+4 y=\sqrt{7}$ touches the ellipse $3 x^{2}+4 y^{2}=1$, then the point of contact is

(a) $\left(\frac{1}{\sqrt{7}}, \frac{1}{\sqrt{7}}\right)$

(b) $\left(\frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}\right)$

(c) $\left(\frac{1}{\sqrt{7}}, \frac{-1}{\sqrt{7}}\right)$

(d) None of these

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Solution : (a)

Let $\mathrm{P}\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ be point of contact the equation of tangent to the ellipse

$\frac{x^{2}}{\frac{1}{3}}+\frac{y^{2}}{\frac{1}{4}}=1$ is $\frac{x _{1}}{\frac{1}{3}}+\frac{y _{1}}{\frac{1}{4}}=1$

$3 \mathrm{xx} _{1}+4 \mathrm{yy} _{1}-1=0……..(1)$

Given that $3 x+4 x-\sqrt{7}=0………(2)$ touches the ellipse

$\therefore$ (1) and (2) are same

By comparing we get

$ \begin{aligned} & \frac{3 x _{1}}{3}=\frac{4 y _{1}}{4}=-\frac{-1}{-\sqrt{7}} \\ & x=\frac{1}{\sqrt{7}}, y _{1}=\frac{1}{\sqrt{7}} \\ & \left(\frac{1}{\sqrt{7}}, \frac{1}{\sqrt{7}}\right) \text { is the point of contact } \end{aligned} $

2. The number of values of c such that the line $y=4 x+c$ touches the curve $\frac{x^{2}}{4}+y^{2}=1$ is

(a) 0

(b) 1

(c) 2

(d) infinite.

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Solution : Given ellipse is $\frac{x^{2}}{4}+\frac{y^{2}}{1}$

$ \mathrm{a}^{2}=4 \quad \mathrm{~b}^{2}=1 $

and a line $y=4 x+c$ is a tangent

$ \begin{aligned} & \mathrm{m}=4 \\ & \begin{aligned} \therefore \mathrm{c} & = \pm \sqrt{\mathrm{a}^{2} \mathrm{~m}^{2}+\mathrm{b}^{2}} \\ & = \pm \sqrt{4 \times 16+1} \\ & = \pm \sqrt{65} \end{aligned} \end{aligned} $

$\therefore$ c has 2 values

$ c=\sqrt{65} \text { or }-\sqrt{65} $

3. If $\sqrt{3} b x+a y=2 a b$ touches the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ then the eccentric angle of the point of contact is

(a) $\frac{\pi}{6}$

(b) $\frac{\pi}{4}$

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

(d) $\frac{\pi}{2}$

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Solution : (a)

Equation of tangent $\frac{x}{a} \frac{\sqrt{3}}{2}+\frac{y}{b} \frac{1}{2}=1……(1)$

and equation of tangent at the point $(a \cos \theta, b \sin \theta)$ is $\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1…..(2)$

comparing (1) & (2) we get

$ \begin{aligned} & \cos \theta=\frac{\sqrt{3}}{2} \text { and } \sin \theta=\frac{1}{2} \\ & \therefore \tan \theta=\frac{1}{\sqrt{3}}=\tan \frac{\pi}{6} \\ & \therefore \theta=\frac{\pi}{6} \end{aligned} $

4. A tangent having slope of $\frac{-4}{3}$ to the ellipse $\frac{x^{2}}{18}+\frac{y^{2}}{32}=1$ intersects the major and minor axes at points $\mathrm{A}$ and $\mathrm{B}$ respectively. If $\mathrm{C}$ is the centre of the ellipses, then the area of the triangle $\mathrm{ABC}$ is

(a) 12 sq.u

(b) 24 sq.u

(c) 36 sq.u

(d) 48 sq.u

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Solution : (b)

Equation of tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is

$ y=m x+\sqrt{a^{2} m^{2}+b^{2}} \quad(b>a) $

Here $\mathrm{m}=\frac{-4}{3}, \mathrm{a}^{2}=18, \mathrm{~b}^{2}=32$

$ \begin{aligned} & y=\frac{-4}{3} x+\sqrt{18 \times \frac{16}{9}+32} \\ & y=\frac{-4}{3} x+8 \end{aligned} $

Then points on the axis where tangents meet are $\mathrm{A}(6,0)$ and $\mathrm{B}(0,8)$

Then area of $\triangle \mathrm{ABC}$ is $\frac{1}{2} \times 6 \times 8=24$ sq.u

5. If the tangents to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ make angle $\alpha$ and $\beta$ with the major axis such that $\tan \alpha+\tan \beta=\lambda$, then the locus of their point intersection is

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

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

(c) $x^{2}-a^{2}=2 \lambda x y$

(d) $\lambda\left(x^{2}-a^{2}\right)=2 x y$

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Solution : (d)

Equation of tangent to the ellipse with slope $m$ is

$ y=m x+\sqrt{a^{2} m^{2}+b^{2}} $

If it is passes through the point $\mathrm{P}(\mathrm{h}, \mathrm{k})$ then

or $(\mathrm{k}-\mathrm{mh})^{2}=\sqrt{\mathrm{a}^{2} \mathrm{~m}^{2}+\mathrm{b}^{2}}$

$ \mathrm{k}=\mathrm{mh}+\sqrt{\mathrm{a}^{2} \mathrm{~m}^{2}+\mathrm{b}^{2}} $

$k^{2}+m^{2} h^{2}-2 m k h=a^{2} m^{2}+b^{2}$

$\mathrm{m}^{2}\left(\mathrm{~h}^{2}-\mathrm{a}^{2}\right)-2 \mathrm{mkh}+\mathrm{k}^{2}-\mathrm{b}^{2}=0$

It is a quadratic in $\mathrm{m}$ having two roots $\mathrm{m} _{1} & \mathrm{~m} _{2}$

$ \mathrm{m} _{1}+\mathrm{m} _{2}=\frac{2 \mathrm{kh}}{\mathrm{h}^{2}-\mathrm{a}^{2}} \text { and } \mathrm{m} _{1} \mathrm{~m} _{2}=\frac{\mathrm{k}^{2}-\mathrm{b}^{2}}{\mathrm{~h}^{2}-\mathrm{a}^{2}} $

Given that $\tan \alpha+\tan \beta=\lambda$

$ \begin{aligned} & \mathrm{m} _{1}+\mathrm{m} _{2}=\lambda \\ & \frac{2 \mathrm{kh}}{\mathrm{h}^{2}-\mathrm{a}^{2}}=\lambda \\ & 2 \mathrm{kh}=\lambda\left(\mathrm{h}^{2}-\mathrm{a}^{2}\right) \end{aligned} $

$\therefore$ locus of point $\mathrm{P}(\mathrm{h}, \mathrm{k})$ is

$ \lambda\left(\mathrm{x}^{2}-\mathrm{a}^{2}\right)=2 \mathrm{xy} $

Practice questions

1. If $\mathrm{P}(\mathrm{x}, \mathrm{y}), \mathrm{F} _{1}(3,0), \mathrm{F} _{2}(-3,0)$ and $16 \mathrm{x}^{2}+25 \mathrm{y}^{2}=400$, then $\mathrm{PF} _{1}+\mathrm{PF} _{2}$ equals

(a) 8

(b) 6

(c) 10

(d) 12

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Answer: (c)

2. The length of the major axis of the ellipse

$(5 x-10)^{2}+(5 y+15)^{2}=\frac{(3 x-4 y+7)^{2}}{4}$ is

(a) 10

(b) $\frac{20}{3}$

(c) $\frac{20}{7}$

(d) 4

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Answer: (b)

3. Angle subtended by common tangents of two ellipses $4(x-4)^{2}+25 y^{2}=100$ and $4(x+1)^{2}+y^{2}=4$ at origin is

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

(b) $\frac{\pi}{4}$

(c) $\frac{\pi}{6}$

(d) $\frac{\pi}{2}$

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Answer: (b)

4. The distance of a point on the ellipse $\frac{x^{2}}{6}+\frac{y^{2}}{2}=1$ from the centre is 2 . Then the eccentric angle of the point is

(a) $\frac{\pi}{4}$

(b) $\frac{3 \pi}{4}$

(c) $\frac{5 \pi}{6}$

(d) $\frac{\pi}{6}$

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Answer: (a, b)

5. If the chord through the points whose eccentric angles are $\theta$ and $\phi$ on the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$ passes through a focus, then the value of $\tan \frac{\theta}{2} \tan \frac{\phi}{2}$ is

(a) $\frac{1}{9}$

(b) $-9$

(c) $\frac{-1}{9}$

(d) $9$

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Answer: (c, d)

6. In an ellipse the distance between its foci is 6 and its minor axis is 8 , the eccentricity of the ellipse is

(a) $\frac{4}{5}$

(b) $\frac{3}{5}$

(c) $\frac{1}{\sqrt{52}}$

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

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Answer: (b)

7. The number of values of $C$ such that the straight line $y=4 x+c$ touches the curve $\frac{x^{2}}{4}+y^{2}=1$, is

(a) $0$

(b) $2$

(c) $1$

(d) $\infty$

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Answer: (b)

8. The line $3 x+5 y=15 \sqrt{2}$ is a tangent to the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$, at a point whose eccentric angle is

(a) $\frac{\pi}{6}$

(b) $\frac{\pi}{4}$

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

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

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Answer: (b)

9. Tangents are drawn to the ellipse $3 x^{2}+5 y^{2}=32$ and $25 x^{2}+9 y^{2}=450$ passing through the point $(3,5)$. The number of such tangents are

(a) $2$

(b) $3$

(c) $4$

(d) $0$

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Answer: (b)

10. Tangents are drawn to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$ at ends of latus rectum. The area of quadrilateral so formed is

(a) $27$

(b) $\frac{27}{2}$

(c) $\frac{27}{4}$

(d) $\frac{27}{55}$

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Answer: (a)

11. An ellipse passes through the point (4,-1) and its axes are along the axes of coordinates. If the line $x+4 y-10=0$ is a tangent to it then its equation is

(a) $\frac{\mathrm{x}^{2}}{100}+\frac{\mathrm{y}^{2}}{5}=1$

(b) $\frac{x^{2}}{8}+\frac{y^{2}}{5 / 4}=1$

(c) $\frac{x^{2}}{20}+\frac{y^{2}}{5}=1$

(d) None of these

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Answer: (b, c)

12. Prove that the line $2 x+3 y=12$ touches the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=2$

13. The tangent at the point $\left(4 \cos \phi, \frac{16}{\sqrt{11}} \sin \phi\right)$ to the ellipse $16 x^{2}+11 y^{2}=256$ is also a tangent to the circle $x^{2}+y^{2}-2 x=15$, find the value of $\phi$.

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Answer: $\pm\frac{\pi}{6}$

14. Find the equations of tangents to the ellipse $9 x^{2}+16 y^{2}=144$ which pass through the point $(2,3)$.

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Answer: $y =3,x + y =5$

15. The angle between pair of tangents drawn to the ellipse $3 x^{2}+2 y^{2}=5$ from the point $(1,2)$ is $\tan ^{1}(12 /$ $\sqrt{5}$ )

16. Prove that the portion of the tangent to the ellipse intercepted between the curve and the directrix subtends a right angle at the corresponding focus.

Linked Comprehension Type.

17. For all real $\mathrm{p}$, the line $2 \mathrm{px}+\mathrm{y} \sqrt{1-\mathrm{p}^{2}}=1$ touches a fixed ellipse whose axes are coordinate axes.

(i). The eccentricity of the ellipse is

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

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

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

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

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Answer: (a)

(ii). The foci of ellipse are

(a) $(0, \pm \sqrt{3})$

(b) $(0, \pm 2 / 3)$

(c) $( \pm \sqrt{3} / 2,0)$

(d) None of these

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Answer: (d)

(iii). The locus of point of intersection of perpendicular tangents is

(a) $\mathrm{x}^{2}+\mathrm{y}^{2}=5 / 4$

(b) $\mathrm{x}^{2}+\mathrm{y}^{2}=3 / 2$

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

(d) None of these

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Answer: (a)

18. $\mathrm{C} _{1}: \mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{r}^{2}$ and $\mathrm{C} _{2}=\frac{\mathrm{x}^{2}}{16}+\frac{\mathrm{y}^{2}}{9}=1$ intersect at four distinct points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}$, Their common tangents form a parallelogram $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime} \mathrm{D}^{\prime}$.

(i). If $\mathrm{ABCD}$ is a square then $\mathrm{r}$ is equal to

(a) $\frac{12}{5} \sqrt{2}$

(b) $\frac{12}{5}$

(c) $\frac{12}{5 \sqrt{5}}$

(d) None of these

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Answer: (a)

(ii). If $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime} \mathrm{D}^{\prime}$ is a square then $\mathrm{r}$ is equal to

(a) $\sqrt{20}$

(b) $\sqrt{12}$

(c) $\sqrt{15}$

(d) None of these

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Answer: (d)

(iii). If $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime} \mathrm{D}^{\prime}$ is a square, then the ratio of area of the circle $\mathrm{C} _{1}$ to the area of the circumcircle of $\Delta \mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$ is

(a) $\frac{9}{16}$

(b) $\frac{3}{4}$

(c) $\frac{1}{2}$

(d) None of these

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Answer: (c)

19. The ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is such that it has the least area but contains the circle $(x-1)^{2}+y^{2}=1$

(i). The eccentricity of the ellipse is

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

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

(c) $\frac{1}{2}$

(d) None of these

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Answer: (a)

(ii). Equation of auxilliary circle of ellipse is

(a) $\mathrm{x}^{2}+\mathrm{y}^{4}=6.5$

(b) $x^{2}+y^{4}=5$

(c) $x^{2}+y^{4}=45$

(d) None of these

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Answer: (c)

(iii). Length of latus rectum of the ellipse is

(a) 2 units

(b) 1 unit

(c) 3 units

(d) 2.5 units

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Answer: (b)

20. The equation of the straight lines joining the foci of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ to the foci of the ellipse $\frac{x^{2}}{24}+\frac{y^{2}}{49}=1$ forms a parallelogram. Then the area of the figure formed by the foci of these two ellipse.

(a) 15

(b) 30

(c) 20

(d) 18

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Answer: (b)