Coordinates from geometric constraints

Find unknown coordinates of a point given geometric constraints like perpendicularity, distance conditions, or lying on a specific line.

32 questions · Moderate -0.0

1.03a Straight lines: equation forms y=mx+c, ax+by+c=0
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Edexcel C1 Q8
9 marks Moderate -0.8
\includegraphics{figure_2} The points \(A(1, 7)\), \(B(20, 7)\) and \(C(p, q)\) form the vertices of a triangle \(ABC\), as shown in Figure 2. The point \(D(8, 2)\) is the mid-point of \(AC\).
  1. Find the value of \(p\) and the value of \(q\). [2]
The line \(l\), which passes through \(D\) and is perpendicular to \(AC\), intersects \(AB\) at \(E\).
  1. Find an equation for \(l\), in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
  2. Find the exact \(x\)-coordinate of \(E\). [2]
Edexcel C1 Q8
12 marks Moderate -0.8
The points \(A(-1, -2)\), \(B(7, 2)\) and \(C(k, 4)\), where \(k\) is a constant, are the vertices of \(\triangle ABC\). Angle \(ABC\) is a right angle.
  1. Find the gradient of \(AB\). [2]
  2. Calculate the value of \(k\). [2]
  3. Show that the length of \(AB\) may be written in the form \(p\sqrt{5}\), where \(p\) is an integer to be found. [3]
  4. Find the exact value of the area of \(\triangle ABC\). [3]
  5. Find an equation for the straight line \(l\) passing through \(B\) and \(C\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [2]
OCR MEI C1 2011 June Q2
2 marks Easy -1.3
A line has gradient 3 and passes through the point \((1, -5)\). The point \((5, k)\) is on this line. Find the value of \(k\). [2]
Edexcel C1 Q6
8 marks Moderate -0.8
The straight line \(l\) passes through the point \(P(-3, 6)\) and the point \(Q(1, -4)\).
  1. Find an equation for \(l\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
The straight line \(m\) has the equation \(2x + ky + 7 = 0\), where \(k\) is a constant. Given that \(l\) and \(m\) are perpendicular,
  1. find the value of \(k\). [4]
Edexcel C2 Q7
10 marks Moderate -0.3
The points \(P\), \(Q\) and \(R\) have coordinates \((-5, 2)\), \((-3, 8)\) and \((9, 4)\) respectively.
  1. Show that \(\angle PQR = 90°\). [4]
Given that \(P\), \(Q\) and \(R\) all lie on circle \(C\),
  1. find the coordinates of the centre of \(C\), [3]
  2. show that the equation of \(C\) can be written in the form $$x^2 + y^2 - 4x - 6y = k,$$ where \(k\) is an integer to be found. [3]
AQA AS Paper 1 2018 June Q5
5 marks Standard +0.3
Point \(C\) has coordinates \((c, 2)\) and point \(D\) has coordinates \((6, d)\). The line \(y + 4x = 11\) is the perpendicular bisector of \(CD\). Find \(c\) and \(d\). [5 marks]
SPS SPS SM Pure 2023 September Q2
6 marks Moderate -0.8
\includegraphics{figure_2} The figure above shows a triangle with vertices at \(A(2,6)\), \(B(11,6)\) and \(C(p,q)\).
  1. Given that the point \(D(6,2)\) is the midpoint of \(AC\), determine the value of \(p\) and the value of \(q\). [2]
The straight line \(l\) passes through \(D\) and is perpendicular to \(AC\). The point \(E\) is the intersection of \(l\) and \(AB\).
  1. Find the coordinates of \(E\). [4]