and C2 touch and find an equation for their common tangent. 6. A circle S, which touches the y-axis, has its centre at the point (1, 2). Find the equation of S. Show that S touches the line 3y- 4x + 3 = 0 and find the coordinates of the point of contact. Find also the equation and the length of the second tangent from the origin to the circles. 7. Given that circles C1: x+y-6x- 4y +9= 0 and C2: x + y-2x - 6y+9 = 0. Find an equation of the circle C3 which passes through the centre of C1 and dhrough the points of intersection of C, and C2. Find also a) An equation of the common chord of C1, C2 and C3 b) Equations for two tangents from the origin to the circle C, which are perpendicular to the common tangent. 8. The equation of two circles C, and C2 are x + y = 4 and C2: x+y-2x 0 respectively a) Show that the circles Ci and C2 touch each other internally and find an equation of the common tangent at the point of contact.

and C2 touch and find an equation for their common tangent. 6. A circle S, which touches the y-axis, has its centre at the point (1, 2). Find the equation of S. Show that S touches the line 3y- 4x + 3 = 0 and find the coordinates of the point of contact. Find also the equation and the length of the second tangent from the origin to the circles. 7. Given that circles C1: x+y-6x- 4y +9= 0 and C2: x + y-2x - 6y+9 = 0. Find an equation of the circle C3 which passes through the centre of C1 and dhrough the points of intersection of C, and C2. Find also a) An equation of the common chord of C1, C2 and C3 b) Equations for two tangents from the origin to the circle C, which are perpendicular to the common tangent. 8. The equation of two circles C, and C2 are x + y = 4 and C2: x+y-2x 0 respectively a) Show that the circles Ci and C2 touch each other internally and find an equation of the common tangent at the point of contact.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

Solve all Q7 explaining detailly each step

CIRCLES
1. Find the coordinates of the center and the length of the radius of the circle:
S: x+y +2x– 4y – 8 = 0. Obtain an equation of the circle S, with center A(3, -4) and radius of
length v13 show that S and S2 touch each other externally and find the coordinates of the point T,
of the contact. Find an equation of the common tangent to the circles at T. Find also equations of
the tangents form the origin to the circle S2.
2. Pand Q are points (1, 2) and (7, 8) respectively. Find an equation of the circ le on PQ as diameter.
Find also an equation of the tangent to this circle at the point P.
V
3. S and S2 are two concentric circles with centre A. Then the radius of S is three times the radius of
S2. Given that the equation of S1 is x
² + y² + 2x – 4y – 31 = 0. Find
%3D
|
An equation of the circle S2
b. An equation of the circle on OA as diameter, where O is the origin.
4. A circle with centre at (2, 3) touches the line 3x – 4y = 5. Find an equation of the circle.
5. Given the two circles C, and C2; C1= x²+y²- 4x-6y+12=0; C2: x+y²– 4x – 8y + l6 = 0 show that C1
and C2 touch and find an equation for their common tangent.
6. A circle S1 which touches the y-axis, has its centre at the point (1, 2). Find the equation of S. Show
that S touches the line 3y – 4x + 3 = 0 and find the coordinates of the point of contact. Find also the
a.
equation and the length of the second tangent from the origin to the circles.
7. Given that circles C1: x+ y- 6x - 4y + 9 = 0 and C2: x + y– 2x – 6y+9 = 0. Find an equation of
the circle C3 which passes through the centre of C and through the points of intersection of C, and
C2. Find also
a) An equation of the common chord of C1, C2 and C3
b) Equations for two tangents from the origin to the circle C1 which are perpendicular to the
common tangent.
%3D
%3D
2
8. The equation of two circles C, and C2 are x+ y = 4 and C2: x+y- 2x 0 respectively
a) Show that the circies C and C2 touch each other internally and find an equation of the common
tangent at the point of contact.
b) Find the equation of the tangents to the circle Ci which are perpendicular to the common tangent
c) Find the area between the two circles
9. The line y = x is a tangent to the circle S at the point (4, 4) on S. The line y = 3x passes through the
centre of the circle. Find an equation of the circle S.
10. Find an equation of the circle which has a segment of the line y = x+ 2 as a diameter and passes
through the point (1, 1) and (2, 3). Find the equations of the two tangents from the origin to the
circle.

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