Standard +0.3 This is a straightforward coordinate geometry problem requiring the equation of a line (y - 5 = -2(x - 3)), then using the distance formula √[(x-3)² + (y-5)²] = 6√5 with the constraint that B lies on the line. Substituting y = -2x + 11 gives a simple quadratic equation. While it involves multiple steps, each technique is standard and the problem structure is typical for C1, making it slightly easier than average.
8 The line \(l\) has gradient - 2 and passes through the point \(A ( 3,5 ) . B\) is a point on the line \(l\) such that the distance \(A B\) is \(6 \sqrt { 5 }\). Find the coordinates of each of the possible points \(B\).
\(B\) lies on \(l\) so has coordinates \((x,\, 11-2x)\)
M1
Attempt to find equation of \(l\) with gradient \(-2\)
\((x-3)^2+(11-2x-5)^2=(6\sqrt{5})^2\)
M1
\((x-3)^2+(y-5)^2=(6\sqrt{5})^2\) o.e. seen
\(5x^2-30x-135=0\)
M1*
Attempts to solve the equations simultaneously to get a quadratic
\(5(x+3)(x-9)=0\)
M1dep
Correct method to solve their quadratic
\(x=-3,\; x=9\)
A1
Both \(x\) values
\(y=17,\; y=-7\)
A1
Both \(y\) values
Alternative method: Use of \((1,2,\sqrt{5})\) triangle with \(-\)ve gradient M1; scaling to \(6\sqrt{5}\) M1; \((3,5)+(6,-12)\) M1; \((9,-7)\) A1; \((3,5)-(6,-12)\) M1; \((-3,17)\) A1
SC: If A0A0, one correct pair of values from correct factorisation www B1. Spotted solutions: each correct pair www B1; checks solutions and justifies only two solutions B2
## Question 8:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $B$ lies on $l$ so has coordinates $(x,\, 11-2x)$ | M1 | Attempt to find equation of $l$ with gradient $-2$ |
| $(x-3)^2+(11-2x-5)^2=(6\sqrt{5})^2$ | M1 | $(x-3)^2+(y-5)^2=(6\sqrt{5})^2$ o.e. seen |
| $5x^2-30x-135=0$ | M1* | Attempts to solve the equations simultaneously to get a quadratic |
| $5(x+3)(x-9)=0$ | M1dep | Correct method to solve their quadratic |
| $x=-3,\; x=9$ | A1 | Both $x$ values |
| $y=17,\; y=-7$ | A1 | Both $y$ values |
**Alternative method:** Use of $(1,2,\sqrt{5})$ triangle with $-$ve gradient **M1**; scaling to $6\sqrt{5}$ **M1**; $(3,5)+(6,-12)$ **M1**; $(9,-7)$ **A1**; $(3,5)-(6,-12)$ **M1**; $(-3,17)$ **A1**
**SC:** If A0A0, one correct pair of values from correct factorisation www **B1**. Spotted solutions: each correct pair www **B1**; checks solutions and justifies only two solutions **B2**
8 The line $l$ has gradient - 2 and passes through the point $A ( 3,5 ) . B$ is a point on the line $l$ such that the distance $A B$ is $6 \sqrt { 5 }$. Find the coordinates of each of the possible points $B$.
\hfill \mbox{\textit{OCR C1 2012 Q8 [6]}}