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Questions and Answers
What determines the number of right-angled triangles that can be inscribed in a circle when the height is greater than the radius?
What determines the number of right-angled triangles that can be inscribed in a circle when the height is greater than the radius?
- There are two right-angled triangles.
- There is one right-angled triangle.
- There are no right-angled triangles. (correct)
- Infinite right-angled triangles can be inscribed.
Which of the following statements about the nature of lines defined by the equation $ax^2 + 2hxy + by^2$ is correct when $h^2 - ab < 0$?
Which of the following statements about the nature of lines defined by the equation $ax^2 + 2hxy + by^2$ is correct when $h^2 - ab < 0$?
- The lines are real and distinct.
- The lines are imaginary. (correct)
- The lines are parallel.
- The lines are coincident.
In determining the area of a parallelogram using area formula $(p_1 p_2) / ext{sin }θ$, what does $θ$ represent?
In determining the area of a parallelogram using area formula $(p_1 p_2) / ext{sin }θ$, what does $θ$ represent?
- The angle between the bases of the parallelogram.
- The angle between the two sides of the parallelogram. (correct)
- The complementary angle to the angle between the sides.
- The angle between the heights of the parallelogram.
When two lines are defined by their respective equations, how does one determine if they are parallel?
When two lines are defined by their respective equations, how does one determine if they are parallel?
What is the condition for the existence of circles that can touch all three lines when the lines are not concurrent and not parallel?
What is the condition for the existence of circles that can touch all three lines when the lines are not concurrent and not parallel?
What is the formula for the area of a triangle given by the equation ax² + 2hxy + by²?
What is the formula for the area of a triangle given by the equation ax² + 2hxy + by²?
In the equation for slopes given by m₁ + m₂ = -2h, what can be concluded about the relationship between the slopes?
In the equation for slopes given by m₁ + m₂ = -2h, what can be concluded about the relationship between the slopes?
What condition must be satisfied for two lines to be perpendicular according to the expressions involving tan θ?
What condition must be satisfied for two lines to be perpendicular according to the expressions involving tan θ?
For an equilateral triangle, which relationship is represented by the parameter tan²α?
For an equilateral triangle, which relationship is represented by the parameter tan²α?
Which statement accurately reflects the relationship between the angles and slopes in right-angled triangles?
Which statement accurately reflects the relationship between the angles and slopes in right-angled triangles?
What is the formula for calculating the acute angle θ between two lines in terms of sine?
What is the formula for calculating the acute angle θ between two lines in terms of sine?
Which statement reflects a relationship involving the centroid of a triangle?
Which statement reflects a relationship involving the centroid of a triangle?
What does the ratio m₁:m₂ = (ab)^(1/n+1) + (ab)^(1/n+1) + 2h signify regarding triangle properties?
What does the ratio m₁:m₂ = (ab)^(1/n+1) + (ab)^(1/n+1) + 2h signify regarding triangle properties?
What is the equation representing the bisectors of the angle formed by two lines denoted as ax² + 2hxy + by² = 0?
What is the equation representing the bisectors of the angle formed by two lines denoted as ax² + 2hxy + by² = 0?
What condition must be satisfied for two lines represented by equations ax² + 2hxy + by² = 0 and lx + my + n = 0 to be parallel?
What condition must be satisfied for two lines represented by equations ax² + 2hxy + by² = 0 and lx + my + n = 0 to be parallel?
Which statement is true about the angular bisectors of the coordinate axes?
Which statement is true about the angular bisectors of the coordinate axes?
When two lines represented by ax² + 2hxy + by² = 0 are perpendicular, which equation must hold true?
When two lines represented by ax² + 2hxy + by² = 0 are perpendicular, which equation must hold true?
For the lines given by the equation ax² + 2hxy + by² = 0, what is the condition for them to have a common line?
For the lines given by the equation ax² + 2hxy + by² = 0, what is the condition for them to have a common line?
What is the equation that indicates two lines from ax² + 2hay + by² = 0 are perpendicular?
What is the equation that indicates two lines from ax² + 2hay + by² = 0 are perpendicular?
Which scenario applies when the lines a₁x² + 2h₁xy + b₁y² = 0 are said to intersect at right angles?
Which scenario applies when the lines a₁x² + 2h₁xy + b₁y² = 0 are said to intersect at right angles?
What is true about the area of a triangle formed by the lines represented as ax² + 2hxy + by² = 0?
What is true about the area of a triangle formed by the lines represented as ax² + 2hxy + by² = 0?
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Study Notes
Slope of Lines
- The equation of a pair of straight lines is given by: y2 - (m1+ m2) xy + m1m2x2 = 0.
- The sum of the slopes of a pair of straight lines is: m1 + m2 = -2h.
- The difference of the slopes of a pair of straight lines is: |m1 - m2| = 2√h2-ab / |b|.
- If the slopes are in the ratio m1:m2, then: (m+n)2 ab = 4 h2 mn.
- If the slopes are in the ratio m1:m2n, then: (ab)1/n+1 + (ab)1/n+1 + 2h = 0.
Angle Between Pair of Lines
- θ is the acute angle between the pair of lines.
- The sine of the angle is given by: sin θ = 2√h2-ab / (a+b).
- The cosine of the angle is given by: cos θ = √(a-b)2 + 4h2 / (a+b).
- The tangent of the angle is given by: tan θ = 2√h2-ab / (a+b).
- If θ = 0°, then x2-ab=0, indicating parallel lines.
- If θ = 90°, then a+b=0, indicating perpendicular lines.
Types of Triangles
- The general equation of a pair of straight lines is: ax2 + 2hxy + by2 = 0.
- The equation of a line is: lx + my + n = 0.
- For an equilateral triangle formed by the pair of straight lines and the line: ax2+2hxy+by2 = (lx+my)2 - 3 (mx - ly)2 .
- For an isosceles triangle formed by the pair of straight lines and the line: h (l2-m2) = (a-b) lm.
- For a right-angled triangle formed by the pair of straight lines and the line:
- a+b=0 (or) a2l2 + 2hlm + b2m2 = 0
- (lx+my)2 - tan2α (mx-ly)2 = 0
- If tan2 α = 3, then the triangle is equilateral.
- If tan2α = 1, then the triangle is a right-angled isosceles triangle.
- If tan2α<1, then the triangle is isosceles and obtuse-angled.
Area of Triangle
- The area of a triangle formed by the pair of straight lines and the line is: h2 √h2-ab / (l2+m2)2.
- For a general equation: (lx+my)2 - tan2α (mx-ly)2 = 0, the area of the triangle is: n2 / (l2+m2)
- For an equilateral triangle: n2 /√3 (l2+m2) = p2 / √3
- p is the perpendicular distance from (0,0) to the line: p = n / √l2+m2.
- For a general equation: (lx+my)2 - k(mx - ly)2 = 0, if k = 3, (lx+my)2 - 3(mx - ly)2 = 0, then the triangle is equilateral.
Centres Related to Triangles
- (α, β) is the centroid of the triangle formed by the pair of straight lines and the given line.
- α = bl-hm / 3(am2-2hlm+bl2)
- β = am-hl / 3(am2-2hlm+bl2)
- (K, Km) is the orthocentre of the triangle formed by the pair of straight lines and the given line.
- K = -n (a+b) / am2 - 2hml + bl2.
- If (l,m) is the orthocentre, then the equation of the 3rd side is: (a+b)(lx+my) = am2 - 2hlm+bl2.
- θ is the angle between the sides of the parallelogram.
- The area of parallelogram is: (p1 p2) / sin θ where p1 and p2 are the lengths of the parallel sides.
- The area of parallelogram is also given by: (c1-c2)(d1-d2) / (a1b2 - a2b1)
- Given the equations: ax1+ by1+c1 = 0, ax1+by1+c2 = 0
- ax2+by2+d1=0, ax2+by2+d2 = 0
- The area of a rhombus is: ½ * d1 * d2, where d1 & d2 are the lengths of the diagonals.
No. of Lines Drawn
- d is the distance from point B.
- If AB = d, then there is 1 line that can be drawn through point A.
- If AB >d, then there are 2 lines that can be drawn through point A.
- If AB < d, then there are 0 lines that can be drawn through point A.
No. of Right Angled Δ in A Circle
- h is the height of the triangle and r is the radius of the circle.
- If h = r, then there are 2 right-angled triangles.
- If h<r, then there is 1 right-angled triangle.
- If h > r, then there are 0 right-angled triangles.
Angular Bisector
- The equations of two lines are: a1x + b1y + c1=0, a2x + b2y + c2 = 0.
- The equation of the angular bisector is given by: (a1x+b1y+c1) / √a12+b12 = (a2x+b2y+c2) / √a22+b22.
- If c1c2 (a1a2+b1b2) < 0, then the origin lies in the acute angle between the lines.
- If c1c2 (a1a2+b1b2) > 0, then the origin lies in the obtuse angle between the lines.
No. of Circles Touching All Three Lines
- If the lines are parallel, then no circle can touch all three lines.
- If the lines are concurrent, then one circle can touch all three lines.
- If two lines are parallel, then two circles can touch all three lines.
- If the lines are not concurrent and not parallel, then four circles can touch all three lines.
Pair of Straight Lines
- The general form of the equation of a pair of straight lines is: ax2 + 2hxy + by2 = 0.
- The separate equations of the lines are:
- ax+(h+ √h2-ab) y = 0
- ax + (h-√h2-ab) y = 0
Nature of Lines
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If h2-ab > 0, then the lines are real and distinct.
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If h2- ab = 0, then the lines coincide.
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If h2-ab < 0, then the lines are imaginary.
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Given the equation of a pair of straight lines, S = ax2 + 2hxy+by2 = 0, and a point (x1,y1) which is the midpoint of the 3rd side of the triangle formed by the lines, then: ax1 + h(xy1+x1y) + by1 = ax12 + 2hx1y1 + by12.
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Given that (x,y) is the centroid of the triangle formed by the pair of straight lines, and S = ax2 + 2 hay + by2 = 0, then:
- S1 = 1/3 S11
- ax1 + hexy1 + yx1) + by1 = 1/3 (ax12 + 2hxiy1+by12)
Intercept of Pair of Lines
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The intercept of the pair of lines ax2 + 2hxy + by2=0, and lx+my+n=0 is: 2 √(h2-ab)(l2+m2)/am2-2him + bl2.
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If aix2 + 2nixy + biy2 = 0, and a2x2+2hz xy + b2y2 = 0, have one line in common, then: (aibz-azbi)2 + 4 (hiaz-hza1) (hibz-habi) = 0.
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If one line is perpendicular to another in the pair of lines, then: (aiaz-bib₂)2 + 4 (hiaz-habı) (hibz-h2a1) = 0.
- If one line is in common and the other lines are perpendicular, then: hia2 b2 - hza1b1 = 1 / √aiaz bibz.
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Given a pair of straight lines ax2 + 2nxy+by2, and a point (x1,y1):
- For parallel lines the equation is: a (x-x1)2 + 2h (x-x1)(y-y1)+b(y-y1)2=0.
- For perpendicular lines the equation is: b (x-x1)2 + 2h(x-x1)(y-y1) + a(y-y1)2 = 0.
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Given a pair of straight lines ax2 + 2hay + by2 = 0 , and a line lx+my+n=0:
- For parallel lines: am2 - 2hlm + bl2 = 0.
- For perpendicular lines: al2 + anim + bm2=0.
Angular Bisectors
- The equation of the pair of angular bisectors is: h (x2-y2) = (a-b)xy.
- The equation of the bisectors of the coordinate axes is: x2- y2 = 0.
- If one of the lines in ax2 + 2hxy + by2 = 0 bisects the angle between the coordinate axes, then: (a+b)2 = 4h2.
- The angle between a pair of angular bisectors is 90° (or) √2.
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