Rajasthan Board RBSE Class 11 Maths Chapter 12 Conic Section Ex 12.6

Question 1.
Prove that liney = x+ touches ellipse 2x² + 3y² = 1. Also find the coordinates of tangent point.
Solution:

Question 2.
Show that line x – 3y – 4 = 0 touches ellipse 3x² + 4y² = 20.
Solution:
Equation of line, x – 3y – 4 =0
⇒ x = 3y + 4
Equation of ellipse,
3x² + 4y² = 20
Putting value of x from (i) in eqn. (ii),
3(3y + 4)² + 4y² = 20
⇒ 3(9y² + 24y + 16) + 4y² – 20 = 0
⇒ 27y² + 72y + 48 + 4y² – 20 = 0
⇒ 31y² + 72y + 28 = 0
Line will touch eqⁿ. (ii) if
(72)² – 4(31 )(28) = 0
∴ (72)² – 4 × 31 × 28 = 5184 – 3472
= 1712 ≠ 0
Thus, given line will not be touch given ellipse.
∴ Given equation is wrong.

Question 3.
For which value of k, the line 3x – 4y = k touches the ellipse 5x² + 4y² = 20.
Solution:
Equation of line,
3x – 4y = k
⇒ 3x – k = 4y
⇒ 4y = 3x – k

Question 4.
Prove that line

touches ellipse

Also find the coordinates of tangent point.
Solution:

Question 5.
Find the condition that line lx + my = n touches the ellipse

Solution:

Question 6.
Find the equations of tangent for ellipse 4x² + 3K² = 5 which makes equation angle of 60° with x-axis. Also find the coordinates of tangent point.
Solution:
Equation of ellipse
4x² + 3y² = 5