Q.Find the conditions for the line lx+my+n=0 to be a tangent to the ellipse a2x2+b2y2=1.
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Coordinate Geometry: Where Algebra Meets Geometry
Imagine you're telling a friend where you left your book in a library. You don't say "near the window" — you say "third shelf, second row, fourth book from the left." You're using numbers to pin down an exact location.
Coordinate geometry does the same thing, but for points on a flat surface. It gives every point a precise address — a pair of numbers — so we can describe shapes, distances, and positions using algebra.
The Big Idea
Before coordinate geometry, geometry was about drawing shapes and proving things with logic alone. Algebra was about numbers and equations. These two worlds seemed separate.
Then René Descartes (a French mathematician) had a simple but revolutionary idea: draw two perpendicular number lines that cross at zero. Now every point on the plane has a unique pair of numbers — its coordinates.
That's it. That's the entire foundation.
The Coordinate System
Take a horizontal line — call it the x-axis. Take a vertical line — call it the y-axis. They cross at a point called the origin, labelled O.
Any point P is located by two numbers:
- Its x-coordinate: how far right (positive) or left (negative) from the origin
- Its y-coordinate: how far up (positive) or down (negative) from the origin
We write this as an ordered pair: (x,y).
The order matters. (3,5) is not the same point as (5,3). The first number is always the horizontal position; the second is always the vertical.
A Concrete Example
Plot the point A(2,3):
- Start at the origin (0,0).
- Move 2 units to the right along the x-axis.
- From there, move 3 units up (parallel to the y-axis).
- Mark the point.
Now plot B(−1,4):
- Start at the origin.
- Move 1 unit left (negative x-direction).
- Move 4 units up.
- Mark the point.
Every point on the plane has exactly one such address. And every pair of numbers corresponds to exactly one point. This one-to-one matching is what makes coordinate geometry powerful.
The Four Quadrants
The axes divide the plane into four regions, called quadrants:
| Quadrant | x-sign | y-sign | Example |
|---|---|---|---|
| I | + | + | (2,3) |
| II | − | + | (−1,4) |
| III | − | − | (−3,−2) |
| IV | + | − | (5,−1) |
Points on the axes themselves (where either coordinate is zero) don't belong to any quadrant.
Why This Matters
Once every point has a number address, we can:
- Calculate distances between points using the Pythagorean theorem
- Find midpoints by averaging coordinates
- Describe lines with equations like y=mx+c
- Solve geometric problems using algebra instead of drawing
The distance between two points (x1,y1) and (x2,y2) is:
d=(x2−x1)2+(y2−y1)2
This is just the Pythagorean theorem in disguise.
The Precise Statement
Coordinate geometry (also called analytic geometry) is the study of geometry using a coordinate system. It establishes a correspondence between:
- Points on a plane and ordered pairs of real numbers
- Geometric figures (lines, circles, curves) and algebraic equations …
Express the line in slope-intercept form and apply the ellipse tangency condition c2=a2m2+b2. …
The line lx+my+n=0 touches the ellipse iff a2l2+b2m2=n2.
Write lx+my+n=0 as y=−mlx−mn (assuming me0), so slope M=−ml and intercept c=−mn.
The condition for y=Mx+c to be a tangent to a2x2+b2y2=1 is
c2=a2M2+b2.
…
Showing the 12 most recent of 23 on this concept.
- CBSE 2026Set ANNUAL1 markMCQQ.Equation of the line passing through the points (−4,3) with slope 21 is(a) x−2y+10=0(b) x−2y−10=0(c) 2x−y+10=0(d) x+2y+10=0
›Reveal solutionSolution
Apply the point-slope equation y−y1=m(x−x1) and simplify to general form.
Point-slope form: y−y1=m(x−x1) with (x1,y1)=(−4,3) and m=21:
y−3=21(x−(−4))=21(x+4)
Multiply both sides by 2: …
- CBSE 2026Set ANNUAL1 markQ.Fill in the blank: Equation of a line making intercepts a and b on the x and y axis respectively is ______.
›Reveal solutionSolution
A line cutting intercepts a and b on the x- and y-axis respectively has equation ax+by=1.
The line passes through (a,0) on the x-axis and (0,b) on the y-axis.
…
- CBSE 2026Set ANNUAL1 markQ.Fill in the blank: The co-ordinates of points in the xy-plane are in the form of ______.
›Reveal solutionSolution
Every point in the Cartesian xy-plane is represented as an ordered pair (x,y).
In the Cartesian coordinate system, two perpendicular number lines (the x-axis and y-axis) intersect at the origin.
…
- CBSE 2025Set ANNUAL1 markMCQQ.Match the column: Column A entry 'y-intercept of the line 2x−3y+6=0' — find the matching value from Column B.(a) 2(b) 8(c) 32(d) 1−tan2x2tanx(e) sin2x(f) 10(g) 20(h) 1+tan2x2tanx(i) 4
›Reveal solutionSolution
Substituting x=0 into the line equation 2x−3y+6=0 gives y=2, the y-intercept.
For the line 2x−3y+6=0, the y-intercept is found by setting x=0:
2(0)−3y+6=0⇒−3y=−6⇒y=2. …
- CBSE 2024Set ANNUAL1 markMCQQ.The equation of the line passing through the points (2,1) and (5,−2) is(a) x+y−3=0(b) x+y+3=0(c) 5x+3y+2=0(d) None of these
›Reveal solutionSolution
Find the slope from the two given points, then use the point-slope form of a line.
Slope through (2,1) and (5,−2):
m=5−2−2−1=3−3=−1
Using point-slope form y−y1=m(x−x1) with point (2,1):
y−1=−1(x−2) …
- CBSE 2024Set ANNUAL1 markMCQQ.If the line (x−y+2)+k(2x+3y+5)=0 is parallel to the line 3x+y=0, then the value of k will be(a) k=1(b) k=74(c) k=47(d) None of these
›Reveal solutionSolution
Expand the family of lines to standard form Ax+By+C=0, write its slope −A/B, and set it equal to the slope of the given parallel line.
Expand (x−y+2)+k(2x+3y+5)=0:
x−y+2+2kx+3ky+5k=0
(1+2k)x+(3k−1)y+(2+5k)=0
The slope of a line Ax+By+C=0 is −A/B, so this line's slope is:
m1=−3k−11+2k
The line 3x+y=0 has slope m2=−3.
…
- CBSE 2023Set ANNUAL1 markMCQQ.The equation of the straight line which passes through the point (4,3) and parallel to the line 3x+4y=12 is(a) 3x+4y=10(b) 3x+4y=24(c) 3x+4y−20=0(d) 3x−4y+24=0
›Reveal solutionSolution
Parallel lines share the same x,y coefficients; substituting the given point fixes the constant as 24.
A line parallel to 3x+4y=12 has the same left-hand-side coefficients, differing only in the constant:
3x+4y=c
…
- CBSE 2023Set ANNUAL1 markQ.Find the equation of the line through (−2,3) with slope −4.
›Reveal solutionSolution
The equation of the line is 4x+y+5=0.
Using point-slope form:
y−3=−4(x−(−2))=−4(x+2) …
- CBSE 2023Set ANNUAL1 markQ.Find the equation of the line through the points (3,−2) and (−1,4).
›Reveal solutionSolution
The equation of the line through (3,−2) and (−1,4) is 3x+2y−5=0.
Slope: m=−1−34−(−2)=−46=−23.
Using point-slope form through (3,−2): …
- CBSE 2023Set ANNUAL1 markMCQQ.Case study: Due to Covid-19 pandemic situation, people are maintaining social distances. Three friends Along, Mharemo and Senti are sitting on the vertices of a triangle whose coordinates are A(3, 1), M(2, -3) and S(-3, 3). The equation of the line AM is(a) 4x+y−11=0(b) 4x−y−11=0(c) 4x+y+11=0(d) 4x−y+11=0
›Reveal solutionSolution
Find the slope of AM, then use point-slope form through A.
A(3,1), M(2,−3).
Slope of AM =2−3−3−1=−1−4=4.
…
- CBSE 2023Set ANNUAL1 markMCQQ.Case study (continued — triangle with vertices A(3,1), M(2,-3), S(-3,3)): The equation of a line passing through A and parallel to SM is(a) 6x+5y−23=0(b) 6x−5y−23=0(c) 6x−5y+23=0(d) 6x+5y+23=0
›Reveal solutionSolution
Find slope of SM, then write the parallel line through A with that same slope.
S(−3,3), M(2,−3).
Slope of SM =2−(−3)−3−3=5−6.
…
- CBSE 2023Set ANNUAL1 markMCQQ.Case study (continued — triangle with vertices A(3,1), M(2,-3), S(-3,3)): Equation of median through A is(a) 2x+7y+1=0(b) 2x−7y+1=0(c) 2x+7y−1=0(d) 2x−7y−1=0
›Reveal solutionSolution
The median from A passes through the midpoint of the opposite side SM.
Midpoint of S(−3,3) and M(2,−3): (2−3+2,23−3)=(−21,0).
Slope of the median (through A(3,1) and this midpoint): −1/2−30−1=−7/2−1=72.
…
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