Coordinate Geometry Problems with Unknown Parameters

Coordinate-geometry parameter problems ask you to turn a geometric condition into an algebraic equation. The unknown may sit inside a coordinate, a line equation or a ratio; the geometry tells you what equation that parameter must satisfy.

1. Translate the condition before solving

• Parallel lines → equal gradients, with special care for vertical lines.
• Perpendicular nonvertical lines → product of gradients −1.
• Midpoint condition → coordinate averages.
• Equal lengths → equal squared distances.
• Collinear points → equal segment gradients or a valid line equation.

2. Parameter from parallelism

Points A(1,2), B(5,k) form a line parallel to y=3x−4. Gradient AB=(k−2)/4. Set this equal to 3:

(k−2)/4=3, so k=14.

3. Parameter from perpendicularity

A(0,1), B(2,k) lies on a line perpendicular to y=−(1/2)x+7. The given gradient is −1/2, so the perpendicular gradient is 2.

(k−1)/2=2, giving k=5.

4. Parameter from a midpoint condition

The midpoint of A(k,4) and B(8,−2) is M(5,1). The y-coordinate already checks: (4−2)/2=1. For x, (k+8)/2=5, so k=2.

5. Parameter from equal distances

Find k if P(k,0) is equidistant from A(1,2) and B(5,2).

Compare squared distances to avoid unnecessary square roots:

(k−1)²+4=(k−5)²+4.

Thus (k−1)²=(k−5)², giving k=3. Geometrically, P lies on the perpendicular bisector of AB.

6. Parameter from collinearity

A(1,3), B(4,9), C(k,15) are collinear. Gradient AB=6/3=2. Therefore gradient BC=(15−9)/(k−4)=2.

6/(k−4)=2, so k−4=3 and k=7.

7. A parameter can create more than one solution

Find k if the distance from P(k,0) to A(0,0) is 4. Then √(k²)=4, so |k|=4 and k=4 or −4.

The geometry explains the pair: there are two points on the horizontal axis four units from the origin.

8. Parameter inside a line equation

Find k if kx+2y−6=0 is parallel to 3x−4y+1=0.

The first line has gradient −k/2. The second has gradient 3/4. Equal gradients give −k/2=3/4, so k=−3/2.

9. Verify the geometry after solving

Substitute the parameter back into the coordinate or line equation and recheck the original geometric condition. This catches algebraic solutions produced from an incorrect translation.

For example, after finding k=14 in the parallelism example, gradient AB=(14−2)/4=3, matching the given line.

10. Common mistakes

• Solving for a coordinate before stating the geometric condition.
• Using equal distances but forgetting two possible positions.
• Using the perpendicular-gradient product rule for a vertical line.
• Equating only x-coordinates in a midpoint condition and ignoring y.
• Accepting a parameter without substituting it back into the geometry.

11. Practice

1. A(2,1), B(6,k) lies on a line parallel to y=2x+5. Find k.
2. A(1,4), B(3,k) lies on a line perpendicular to y=x−2. Find k.
3. The midpoint of (k,8) and (10,2) is (6,5). Find k.
4. Find k if P(k,0) is 5 units from the origin.
5. A(0,0), B(3,6), C(k,10) are collinear. Find k.

12. Answers

1. (k−1)/4=2, so k=9.
2. Perpendicular gradient −1. (k−4)/2=−1, so k=2.
3. (k+10)/2=6, so k=2.
4. k=5 or −5.
5. Gradient AB=2. 10/k=2, so k=5.

Continue with Parallel and Perpendicular Lines, Distance, Midpoint and Section Formulae, or return to the Additional Mathematics Hub.