Suggested languages for you:

Americas

Europe

Problem 23

Find the equation of the pair of tangents from the origin to the circle \(x^{2}+y^{2}+2 g x+2 f y+\lambda^{2}=0\), and show that their intercept on the line \(y=h\) iss \(\frac{2 h \lambda}{\lambda^{2}-g^{2}}\) times the radius of the circle.

Expert verified

The equation of the pair of tangents from the origin to the circle is \(-x^2 - y^2 + 2gx + 2fy + g^2 + f^2 - \lambda^2 = 0\). The intercept on the line \(y=h\) is \(\frac{2 h \lambda}{\lambda^{2}-g^{2}}\) times the radius of the circle.

What do you think about this solution?

We value your feedback to improve our textbook solutions.

- Access over 3 million high quality textbook solutions
- Access our popular flashcard, quiz, mock-exam and notes features
- Access our smart AI features to upgrade your learning

Chapter 1

The circle \(x^{2}+y^{2}-6 x-6 y+9=0\) is rolled on the \(x\)-axis in the positive direction through one complete revolution. Find the equation of the circle in the new position.

Chapter 1

\- IHtet by \(\pi / 3\). Prove that the length of the common chord of the two circles \(x^{2}+y^{2}=a^{2}\) and \((x-c)^{2}+y^{2}=b^{2}\) is $$ \frac{1}{c} \sqrt{((a+b+c)(a-b+c)(a+b-c)(-a+b+c)\\}} $$

Chapter 1

One vertex of triangle is \((0,0)\) and another moves along the circumference of the circle \((x-d)^{2}+y^{2}=a^{2}\). Prove that the locus of the remaining vertex is $$ \frac{\sin ^{2} C}{\sin ^{2} B}\left(x^{2}+y^{2}\right)-2 \frac{\sin C}{\sin B} d(x \cos A+y \sin A) d^{2}-a^{2}=0 $$ where \(A, B\) and \(C\) are the angles of the triangle respectively.

Chapter 1

Find the locus of the point of intersection of tangents to the circle $x=a \cos \theta, y=a \sin \theta\( at points whose parametric angles differ by \)\pi / 2$

Chapter 1

Two consecutive vertices of a rectangle of area 10 unit \(^{2}\) are \((1,3)\) and \((-2,-1)\). Other two vertices are: (a) $\left(-\frac{3}{5}, \frac{21}{5}\right),\left(-\frac{18}{5}, \frac{1}{5}\right)$ (b) $\left(-\frac{3}{5}, \frac{21}{5}\right),\left(-\frac{11}{5},-\frac{2}{5}\right)$ (c) $\left(-\frac{2}{5},-\frac{11}{5}\right),\left(\frac{13}{5}, \frac{9}{5}\right)$ (d) $\left(\frac{13}{5}, \frac{9}{5}\right),\left(-\frac{18}{5}, \frac{1}{5}\right)$

The first learning app that truly has everything you need to ace your exams in one place.

- Flashcards & Quizzes
- AI Study Assistant
- Smart Note-Taking
- Mock-Exams
- Study Planner