Skip to main content

Circles

Circles and tangents

Question
If from any point on the circle x2 + y2 +2gx + 2fy + c = 0, tangents are drawn to the circle x2 + y2 + 2gx + 2fy + csin2α + (g2 + f2)cos2 α = 0, then angle between the tangents is 

Concepts used
- Circle with equation x2 + y2 + 2gx + 2fy + c = 0, has center at (-g,-f) and radius √(g2+f2-c),
√ (g2+f2-c) 
- sin2 + cos2 = 1

Answer
Rearrange the equation of circles to find the centers and radii
Circle 1 - x2 + y2 +2gx + 2fy + c = 0
x2 + y2 +2gx + 2fy + c = 0
x2 + y2 +2gx + 2fy + g2 + f2 = g2 + f2 - c 
(x+g)2 + (x+f)2 = (√(g2+f2-c))2

Circle 2 - x2 + y2 + 2gx + 2fy + csin2α + (g2 + f2)cos2 α = 0
x2 + y2 + 2gx + 2fy + g2 + f2 = g2 + f2 - csin2α - (g2 + f2)cos2 α 
(x+g)2 + (x+f)2 = (√(g2 + f2 - csin2α - (g2 + f2)cos2 α ))2

We find that the circles share a common center and differ only in their radii.


From the inherent property of tangents Circle 1 is bigger than Circle 2. It is required to find <

Fact:
If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of themeasures of the arcs intercepted by the angle and its vertical angle.

If a tangent and a secant intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. 

If two tangents intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

If two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

Derivation
Labels:

Popular posts from this blog

The Da Vinci Code trip

Chapter I of the novel begins with Robert Langdon asleep in the Ritz Hotel. Our trip begins with ourselves just outside the Ritz Hotel. One of the posh hotels with a lot of high end luxury cars parked outside. Ritz Hotel, Paris Sitting up now, Langdon frowned at his bedside Guest Relations Handbook, whose cover boasted:     SLEEP LIKE A BABY IN THE CITY OF LIGHTS. SLUMBER AT THE PARIS RITZ. In the beginning on his way to Louvre - The crisp April air whipped through the open window of the Citroën ZX as it skimmed south past the Opera House and crossed Place Vendôme . As the Citroën accelerated southward across the city, the illuminated profile of the Eiffel Tower appeared, shooting skyward in the distance to the right. Symbologists often remarked that France—a country renowned for machismo, womanizing, and diminutive insecure leaders like Napoleon and Pepin the Short—could not have chosen a more apt national emblem than a thous...
Read more

ISRO Mechanical Strength of Materials Questions, Answers and Explanation

Strength of Materials Simple Stresses and Strains 2017.2.9.  Which of the following is true for ductile materials? a) Engineering stress – strain curve cannot have negative slope b) Most applicable failure theory is maximum principal stress theory c) Ultimate strain is the strain at ultimate stress d) Strain hardening is represented by a negative slope in engineering stress strain curve Answer c)  Ultimate strain is the strain at ultimate stress Explanation  Principal stress theory is applicable for brittle materials. Elastic Constants 2017.2. 2.  For isotropic materials, the modulus of Elasticity in tension and shear (E and G) are related to the Poisson's ratio (&nu) as follows a) E = G/(2(1+ν)) b) G = E/(2(1+ν)) c) G = E/(2(1-ν)) d) E = G/(2(1-ν)) Answer b) G = E/(2(1+ν)) Explanation Derivation of relation between Modulus of elasticity in tension and shear Torsion 2017.2.76.  A 3m l...
Read more

ISRO 2017 May Mechanical Questions, Answers and Explanation

ISRO 2017 May Mechanical Questions, Answers and Explanation 1. A solid rod of 12mm diameter was tested for tensile strength with the gauge length of 50mm. Final length = 80mm; Final diameter = 4mm; Yield load = 1130N. What is the nearest yield stress and % reduction in area respectively? a) 10MPa and 10% b) 90MPa and 90% c) 10MPa and 90% d) 90MPa and 10% Answer c) 10MPa and 90% Explanation Yield stress = Yield load/Area at yield = 1130/(π (4x10 -3 ) 2 /4) =  4520x10 6 /(16π) % reduction in area = Change in area*100/Original area 2. When water at 0C is heated, its specific volume under standard atmosphere a) First increases then decreases b) First decreases then increases c) Increases steadily d) Decreases steadily Answer b) First decreases then increases Explanation The density of water is highest at 4C. That is the specific volume is minimum for water at 4C. 3. A spherical joint has ....... degree (s) of freedom. a) One b) Two c) Three d) Four ...
Read more