Find the condition for the curves x 2 a 2 - y 2 b 2 = 1 and x y = c 2 to intersect orthogonally.

Given equation of curve be x 2 a 2 - y 2 b 2 = 1 . . . . . . . . . . . . . . . . . ( 1 ) and x y = c 2 . . . . . . . . . . . . . . . . . ( 2 ) Let ( x 1 , y 1 ) be the point of intersection of two given curves ( 1 ) and ( 2 ) Differentiate ( 1 ) with respect to x , we get 2 x a 2 - 2 y b 2 d y d x = 0 ∴ d y d x = b 2 x a 2 y Thus, m 1 = ( d y d x ) ( x 1 , y 1 ) = b 2 x 1 a 2 y 1 Differentiate ( 2 ) with respect to x , we get x d y d x + y = 0 ⇒ d y d x = - y x m 2 = ( d y d x ) ( x 1 , y 1 ) = - y 1 x 1 Now two curves intersect orthogonally iff m 1 m 2 = - 1 ⇒ ( b 2 x 1 a 2 y 1 ) ( - y 1 x 1 ) = - 1 ⇒ b 2 = a 2 This is the required condition to intersect both curves orthogonally.

E M B I B E

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>CHAPTER TEST

HARD

12th ICSE

IMPORTANT

Earn 100

The radius of a circular blot of ink is increasing at $0.5$ cm per minute. Find the rate of increase of the area when the radius is equal to $5$ cm. Find also the approximate change in the area when the radius increases from $5$ cm to $5.001$ cm.

Important Questions on Applications of Derivatives

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>CHAPTER TEST

A balloon in the form of a right circular cone surmounted by a hemi-sphere, having a diameter equal to the height of the cone, is being inflated by pumping air into it. How fast is its volume changing with respect to its total height

h

, when

h = 9

cm.

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>CHAPTER TEST

Find the equation of the normal to the curve

2 y + x^{2} = 3

at the point

(1, 1)

.

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>CHAPTER TEST

Find the equation of the tangent to the curve $y = x^{3} - x^{2}$ at the point (2,4). Also find the equation of tangents to the curve which are parallel to x-axis.

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>CHAPTER TEST

Find the equation of the tangent to the curve $y = (x^{3} - 1) (x - 2)$ at the points where the curve cuts the x-axis.

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>CHAPTER TEST

Find the equation of the tangent to the curve $y = 3 x^{3} + 4 x^{2} + 2 x$ at the point whose abscissa is $- 1$ . Also find the coordinates of another point on the curve at which the tangent is parallel to that already obtained.