CAIE A-Level Mathematics AS 1.3.5 Graph Intersections Questions

Practise treating graph intersections as simultaneous solutions, eliminating a variable and using roots or discriminants to decide whether curves meet, touch or remain separate.

Exam points

solve coordinate geometry problems
apply algebraic coordinate methods
interpret geometric solutions

[Maximum number: 6]

The straight line y=x+5 meets the curve $2 x^{2}+3 y^{2}=k$ at a single point P.

(a)

Find the value of the constant k.

[ 4 ]

Attempt substitution for y in quadratic equation

*M1

Or substitution for $x \ldots$

Obtain $5 x^{2}+30 x+75-k[=0]$ or $5 y^{2}-20 y+50-k[=0]$
OE e.g. $x^{2}+6 x+15-\frac{k}{5}$ (all terms gathered together).

Use $b^{2}-4 a c=0$ with their a, b and c
${ }^{\prime}=0^{\prime}$ may be implied in subsequent working or the
answer.

Obtain 900-20(75-k)=0 or equivalent and hence k=30
... obtaining 400-20(50-k)=0 and k=30.

(b)

Find the coordinates of P.

[ 2 ]

Substitute their value of k in equation from part (a) and attempt solution
Expect $5 x^{2}+30 x+45[=0]$ or $5 y^{2}-20 y+20[=0]$ .

Obtain coordinates (-3, 2)
SC B1 only (-3, 2) without attempt at quadratic solution.

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CAIE A-Level Mathematics AS 1.3.5 Graph Intersections Questions question 1

Question (a)

Question (b)