Determine whether the line x + 3 y 2 = 9 is a tangent to the ellipse x 2 9 + y 2 4 = 1 . If so, find the co-ordinates of the point of contact.

The given equation of the ellipse is x 2 9 + y 2 4 = 1 . By comparing x 2 9 + y 2 4 = 1 with the standard equation of an ellipse, we have a = 3 , b = 2 . The given equation of the line is x + 3 2 y = 9 and by comparing it with y = m x + c , we get m = - 1 3 2 and c = 9 3 2 ⇒ 3 2 . As we know that the line y = m x + c is tangent to the ellipse x 2 9 + y 2 4 = 1 if c = ± a 2 m 2 + b 2 . By checking the condition for tangency, At LHS , c = 3 2 ⇒ c 2 = 9 2 At RHS , a 2 m 2 + b 2 = 3 2 · - 1 3 2 2 + 2 2 ⇒ a 2 m 2 + b 2 = 9 18 + 4 ⇒ a 2 m 2 + b 2 = 9 2 Here, LHS = RHS which depicts that the line x + 3 2 y = 9 is a tangent to the ellipse x 2 9 + y 2 4 = 1 . Now, the coordinates of the point of contact x 1 , y 1 would be x 1 = - 9 · - 1 3 2 3 2 and y 1 = 4 3 2 . Thus, the point of contact is 1 , 4 2 3 .

E M B I B E

Mathematics and Statistics (Arts & Science) Part 1>Chapter 7 - Conic Sections>EXERCISE 7.1>Q 19

MEDIUM

11th Maharashtra Board

IMPORTANT

Earn 100

The tower of a bridge, hung in the form of a parabola have their tops $30$ meters above the roadway and are $200$ meters apart. If the cable is $5$ meters above the road way at the centre of the bridge, find the length of the vertical supporting cable from the centre.

Important Questions on Conic Sections

MEDIUM

11th Maharashtra Board

IMPORTANT

Mathematics and Statistics (Arts & Science) Part 1>Chapter 7 - Conic Sections>EXERCISE 7.1>Q 20

A circle whose centre is

(4, - 1)

, passes through the focus of the parabola

x^{2} + 16 y = 0

. Show that the circle touches the directrix of the parabola.

HARD

11th Maharashtra Board

IMPORTANT

Mathematics and Statistics (Arts & Science) Part 1>Chapter 7 - Conic Sections>EXERCISE 7.2>Q 1

Find the lengths of the principal axes, coordinates of the foci, equations of directrices, length of the latus rectum, distance between foci, and distance between directrices of the ellipse:

$x^{2} / 25 + y^{2} / 9 = 1$

HARD

11th Maharashtra Board

IMPORTANT

Mathematics and Statistics (Arts & Science) Part 1>Chapter 7 - Conic Sections>EXERCISE 7.2>Q 2

Find the equation of the ellipse in standard form if the distance between its directrix is $10$ and which passes through $(- \sqrt{5}, 2)$ .

HARD

11th Maharashtra Board

IMPORTANT

Mathematics and Statistics (Arts & Science) Part 1>Chapter 7 - Conic Sections>EXERCISE 7.2>Q 2

Find the equation of the ellipse in standard form if eccentricity is $2 / 3$ and passes through $(2, - 5 / 3)$ .

EASY

11th Maharashtra Board

IMPORTANT

Mathematics and Statistics (Arts & Science) Part 1>Chapter 7 - Conic Sections>EXERCISE 7.2>Q 3

If an eccentricity of an ellipse is 2/3 and it passes through (2, −5/3) such that its equation is $\frac{x^{2}}{m} + \frac{y^{2}}{n} = 1$ then find the value of $m + n$ .