Find the length of the chord joining the points in which the straight line x a + y b = 1 meets the circle x 2 + y 2 = r 2 .

The equation of the circle is given as, x 2 + y 2 = r 2 … 1 The circle has the center at ( 0 , 0 ) and a radius r . The equation of the straight line meeting the circle at A and B is given as, x a + y b = 1 … 2 (assume A B ) Draw C N perpendicular from the center on the line. So, N will be the midpoint of A B . N B = 1 2 A B … 3 From the right-angled triangle C N B , we get, N B 2 = C B 2 - C N 2 … 4 The length of the perpendicular from 0 , 0 on A B can be given as, C N = 0 + 0 - a b a 2 + b 2 And, C B is the radius of the circle. Substituting the above values in equation 4 , we get, N B 2 = r 2 - a 2 b 2 a 2 + b 2 N B = r 2 - a 2 b 2 a 2 + b 2 A B = 2 N B A B = 2 r 2 - a 2 b 2 a 2 + b 2

Find the equation of the circle which touches the axes of the coordinates and also the line x a + y b = 1 , the center being in the positive quadrant.

The equation of the line is given as, x a + y b = 1 b x + a y - a b = 0 … i Since the circle is touching both the axes, the center will be h , h and its radius will be h . Since, the circle touches the line i , the length of the perpendicular from the center to the line must be equal to h . Hence, we get, h = b h + a h - a b a 2 + b 2 … ii Squaring equation ii , we get, h 2 b 2 + h 2 a 2 + a 2 b 2 + 2 a b h 2 - 2 h a b 2 + b a 2 = h 2 a 2 + b 2 h = a + b ± a 2 + b 2 2 … iii Hence, the equation of the circle can be given as, x - h 2 + y - h 2 = h 2 x 2 + y 2 - 2 h x - 2 h y + h 2 = 0 (where the value of h is given in equation iii )

E M B I B E

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 6 - The Circle>EXAMPLES XVIII>Q 11

MEDIUM

JEE Advanced

IMPORTANT

Earn 100

Find the equation of the tangent to the circle $x^{2} + y^{2} = a^{2}$ which is perpendicular to the straight line $y = m x + c$ .

Important Questions on The Circle

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 6 - The Circle>EXAMPLES XVIII>Q 11

Find the equation of the tangent to the circle $x^{2} + y^{2} = a^{2}$ which passes through the point $(b, 0)$ .

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 6 - The Circle>EXAMPLES XVIII>Q 11

Find the equation of the tangent to the circle $x^{2} + y^{2} = a^{2}$ which makes a triangle with the axes, where the area of the triangle is $a^{2}$ .