7. Coordinate Geometry
рдЗрд╕ рдкреЗрдЬ рдкрд░ рдЖрдкрдХреЛ 7. Coordinate Geometry рдХреЗ рд╕рднреА рдорд╣рддреНрд╡рдкреВрд░реНрдг рдмрд╣реБрд╡рд┐рдХрд▓реНрдкреАрдп рдкреНрд░рд╢реНрди (MCQ), рдкрд░реАрдХреНрд╖рд╛ рдЙрдкрдпреЛрдЧреА рдиреЛрдЯреНрд╕ рдФрд░ рдСрдирд▓рд╛рдЗрди рдХреНрд╡рд┐рдЬрд╝ рдорд┐рд▓реЗрдВрдЧреЗред рдЕрдкрдиреЗ рдЬреНрдЮрд╛рди рдХрд╛ рдкрд░реАрдХреНрд╖рдг рдХрд░рдиреЗ рдФрд░ рдмреЛрд░реНрдб/рдкреНрд░рддрд┐рдпреЛрдЧреА рдкрд░реАрдХреНрд╖рд╛ рдореЗрдВ рд╢рдд-рдкреНрд░рддрд┐рд╢рдд рдЕрдВрдХ рдкреНрд░рд╛рдкреНрдд рдХрд░рдиреЗ рдХреЗ рд▓рд┐рдП рдиреАрдЪреЗ рджрд┐рдП рдЧрдП рдореЙрдХ рдЯреЗрд╕реНрдЯ рдореЗрдВ рдЕрд╡рд╢реНрдп рднрд╛рдЧ рд▓реЗрдВред
ЁЯСЙ рдЕрдзреНрдпрд╛рдп рдХрд╛ рд╕рд╛рд░рд╛рдВрд╢ (Summary)
Dear students, welcome to Chapter 7 of Class 10 Mathematics, Coordinate Geometry. In Class 9, you successfully learned how to accurately locate the exact position of a distinct point on a 2-dimensional plane using specific coordinates. You already know that the perpendicular distance of a specific point strictly from the y-axis is officially called its x-coordinate, or abscissa, and the perpendicular distance strictly from the x-axis is perfectly known as its y-coordinate, or ordinate. The exact coordinates of any given point naturally lying on the x-axis are strictly of the mathematical form (x, 0), and any specific point logically lying on the y-axis is strictly of the form (0, y). In this crucial chapter, we will continuously build upon these foundational algebraic concepts to mathematically calculate the exact numerical distance strictly between 2 given points, seamlessly find the precise coordinates of a specific point that geometrically divides a linear line segment in a strictly given ratio, and deeply explore related practical applications. This beautiful branch of higher mathematics perfectly bridges the massive gap strictly between pure algebra and pure geometry, directly allowing us to rapidly solve highly complex geometrical problems utilizing simple algebraic equations.
Let us carefully consider a highly relatable real-life example to truly understand this. Imagine you are actively using a digital GPS navigation system on your modern smartphone to accurately find the absolute shortest driving distance perfectly between your home and your school. The advanced GPS strictly uses a massive global coordinate system, essentially horizontal latitude and vertical longitude, to accurately pinpoint exact geographical locations. By systematically applying advanced mathematical principles fundamentally similar to the standard Distance Formula, the software rapidly calculates the precise numerical distance strictly between these 2 locations. The standard Distance Formula is mathematically given by the absolute square root of exactly (x2 - x1)^2 + (y2 - y1)^2. It is directly and logically derived strictly from the famous Pythagoras Theorem. If you specifically need to quickly find the precise distance of any mathematical point (x, y) directly from the fixed origin (0, 0), the complex formula beautifully and instantly simplifies strictly to the square root of exactly x^2 + y^2.
Another extremely powerful and highly practical mathematical tool we will thoroughly master today is the fundamental Section Formula. Suppose a heavy optical telephone cable needs to be laid perfectly straight strictly between 2 major cities, and a large signal relay tower must be exactly constructed such that it logically divides the total linear distance precisely in the numerical ratio 2:1. The powerful Section Formula directly allows us to quickly calculate the exact geographical coordinate points required for correctly constructing this specific tower. If a distinct point P(x, y) logically internally divides the straight line segment directly joining points A(x1, y1) and B(x2, y2) exactly in the numerical ratio m1:m2, then the precise mathematical coordinates of P are perfectly given by strictly ((m1x2 + m2x1)/(m1 + m2), (m1y2 + m2y1)/(m1 + m2)). A very special and frequently utilized geometrical case of this exact formula is the well-known Midpoint Formula, which we logically obtain when the dividing numerical ratio is perfectly 1:1.
Mastering the core principles of Coordinate Geometry is absolutely essential for consistently excelling in your upcoming NCERT and RBSE board exams, as well as strictly for your future higher studies in advanced mechanical physics, digital computer graphics, and structural civil engineering. Vigorously practice strictly applying these mathematical formulas to completely diverse word problems.
рд╡рд┐рд╢реЗрд╖рддрд╛: рдпрд╣ рдПрдХ 3-Level рдЖрдзрд╛рд░рд┐рдд рдпреВрдиреАрдХ рдХреНрд╡рд┐рдЬрд╝ рд╣реИред рдЕрдЧрд▓реЗ рд╕реНрддрд░ (Level 2 рдФрд░ 3) рдХреЛ рдЕрдирд▓реЙрдХ рдХрд░рдиреЗ рдХреЗ рд▓рд┐рдП рдЫрд╛рддреНрд░реЛрдВ рдХреЛ рд╡рд░реНрддрдорд╛рди рд╕реНрддрд░ рдореЗрдВ рдХрдо рд╕реЗ рдХрдо 70% рдЕрдВрдХ рд╕реНрдХреЛрд░ рдХрд░рдиреЗ рд╣реЛрдВрдЧреЗ, рдЬреЛ рдЙрдирдХреА рдмреЛрд░реНрдб рдкрд░реАрдХреНрд╖рд╛ рдХреА рддреИрдпрд╛рд░реА рдХреЛ 100% рдордЬрдмреВрдд рдмрдирд╛рддрд╛ рд╣реИред
рдорд╣рддреНрд╡рдкреВрд░реНрдг рдкреНрд░рд╢реНрди (Important Questions)
PDF рдбрд╛рдЙрдирд▓реЛрдб рдХрд░реЗрдВрдЕрдзреНрдпрд╛рдп рдХреЗ рд╕рднреА рдкреНрд░рд╢реНрди (Revision Notes)
*рдиреЛрдЯ: рдпрд╣ рд╕реЗрдХреНрд╢рди рд░рд┐рд╡реАрдЬрди рдХреЗ рд▓рд┐рдП рд╣реИред рдЕрдкрдирд╛ рдЬреНрдЮрд╛рди рдкрд░рдЦрдиреЗ рдХреЗ рд▓рд┐рдП рдКрдкрд░ рджрд┐рдП рдЧрдП рдХреНрд╡рд┐рдЬрд╝ рдореЗрдВ рднрд╛рдЧ рд▓реЗрдВред
-
What are the exact mathematical coordinates of the origin point?
A) (1, 1) | B) (0, 1) | C) (1, 0) | D) (0, 0) -
Find the exact numerical distance of the specific point P(3, 4) strictly from the origin.
A) 3 | B) 4 | C) 5 | D) 7 -
Any general mathematical point logically lying perfectly on the x-axis is strictly of the form:
A) (0, y) | B) (x, y) | C) (x, 0) | D) (x, x) -
What is the precise numerical distance of the point P(2, 3) strictly from the x-axis?
A) 2 | B) 3 | C) 5 | D) 1 -
What is the exact mathematical midpoint of the line segment joining (2, 4) and (6, 8)?
A) (4, 6) | B) (6, 4) | C) (8, 12) | D) (4, 2) -
In exactly which mathematical quadrant does the point (-3, 4) strictly lie?
A) Quadrant 1 | B) Quadrant 2 | C) Quadrant 3 | D) Quadrant 4 -
Calculate the exact numerical distance perfectly between the points (2, 3) and (4, 1).
A) 2 | B) 2sqrt(2) | C) sqrt(10) | D) 4 -
What is the precise numerical distance of the point Q(5, -2) strictly from the y-axis?
A) -2 | B) 2 | C) 5 | D) 3 -
If exactly the mathematical ordinate is 0, on strictly which axis does the point logically lie?
A) x-axis | B) y-axis | C) Both axes | D) Neither axis -
Find the exact mathematical midpoint of the line segment strictly joining (-2, 8) and (-6, -4).
A) (-4, 2) | B) (4, -2) | C) (-8, 4) | D) (2, 6) -
What is the exact mathematical distance strictly between the specific points (a, b) and (-a, -b)?
A) 0 | B) a^2 + b^2 | C) sqrt(a^2 + b^2) | D) 2sqrt(a^2 + b^2) -
Find the exact coordinates dividing the line strictly joining (4, -3) and (8, 5) internally exactly in 3:1.
A) (7, 3) | B) (3, 7) | C) (5, 1) | D) (6, 2) -
If point (x, y) is perfectly equidistant strictly from points (7, 1) and (3, 5), which mathematical relation is exact?
A) x + y = 2 | B) x - y = 2 | C) x + y = 1 | D) x - y = 1 -
Find the exact numerical ratio in which the x-axis strictly divides the segment joining (1, -5) and (-4, 5).
A) 1:1 | B) 1:2 | C) 2:1 | D) 3:2 -
If points (1, 2), (4, y), (x, 6) and (3, 5) form a parallelogram, find exactly x and y.
A) x=3 y=6 | B) x=6 y=3 | C) x=5 y=4 | D) x=4 y=5 -
Calculate the exact numerical perimeter of a complete triangle with specific vertices (0, 4), (0, 0) and (3, 0).
A) 7 | B) 10 | C) 12 | D) 14 -
Find the specific point strictly on the x-axis that is perfectly equidistant from precisely (2, -5) and (-2, 9).
A) (7, 0) | B) (-7, 0) | C) (0, 7) | D) (0, -7) -
In exactly what specific ratio does the y-axis completely divide the line strictly joining (5, -6) and (-1, -4)?
A) 1:5 | B) 5:1 | C) 2:3 | D) 3:2 -
Calculate the exact numerical distance perfectly between the specific points A(0, 6) and B(0, -2).
A) 4 | B) 6 | C) 8 | D) 10 -
Find the exact mathematical coordinates perfectly dividing the join of A(-1, 7) and B(4, -3) internally in exactly 2:3.
A) (1, 3) | B) (3, 1) | C) (2, 4) | D) (4, 2) -
Find the exact numerical area of the complete rhombus if its mathematical vertices are precisely (3, 0), (4, 5), (-1, 4) and (-2, -1).
A) 12 | B) 24 | C) 36 | D) 48 -
Determine the exact mathematical ratio strictly in which the line 2x + y - 4 = 0 perfectly divides the segment joining A(2, -2) and B(3, 7).
A) 2:9 | B) 9:2 | C) 1:3 | D) 3:1 -
If the specific mathematical points (a, 0), (0, b) and (1, 1) are completely collinear, what is the exact numerical value of 1/a + 1/b?
A) -1 | B) 0 | C) 1 | D) 2 -
Find the exact mathematical coordinates of the centroid of a triangle specifically with vertices (0, 6), (8, 12) and (8, 0).
A) (16/3, 6) | B) (8, 6) | C) (16, 18) | D) (16/3, 18/3) -
Find exactly 1 of the specific coordinates of the points of perfect trisection of the segment strictly joining (4, -1) and (-2, -3).
A) (2, -5/3) | B) (1, -2) | C) (0, -5/3) | D) (2, 1/3) -
What are the precise mathematical coordinates strictly of the center of a circle perfectly passing through (6, -6), (3, -7) and (3, 3)?
A) (3, -2) | B) (3, 2) | C) (-3, 2) | D) (-3, -2) -
If P(x, y) is perfectly any mathematical point logically on the exact line strictly joining A(a, 0) and B(0, b), which relation strictly holds?
A) x/a - y/b = 1 | B) x/a + y/b = 1 | C) x/b + y/a = 1 | D) ax + by = 1 -
The linear segment strictly joining points (1, 2) and (-2, 1) is completely divided mathematically by line 3x + 4y = 7 exactly in what ratio?
A) 3:4 | B) 4:3 | C) 4:9 | D) 9:4 -
What is the exact numerical length strictly of the geometric median through vertex A of a complete triangle with vertices A(-1, 3), B(1, -1) and C(5, 1)?
A) 3 | B) 4 | C) 5 | D) 6 -
If exactly the specific points A(6, 1), B(8, 2), C(9, 4) and D(p, 3) geometrically form a perfect parallelogram, find exactly p.
A) 5 | B) 6 | C) 7 | D) 8
0
0
0%
рдЕрдкрдирд╛ рд╕реБрдЭрд╛рд╡ рджреЗрдВ (Feedback)
рд╡реЗрдмрд╕рд╛рдЗрдЯ рдХреЛ рдФрд░ рдмреЗрд╣рддрд░ рдмрдирд╛рдиреЗ рдХреЗ рд▓рд┐рдП рдЕрдкрдиреЗ рд╡рд┐рдЪрд╛рд░ рд╣рдореЗрдВ рднреЗрдЬреЗрдВред