## What are we learning ?

**Competency in Focus:** Coordinate Geometry

This problem from B.Stat. (Hons.) based on coordinate geometry Admission Test 2005 – Objective Problem 5 is based nature of curve.

## First look at the knowledge graph:-

## Next understand the problem

The equation $x(x+3)=y(y-1)-2$ represents

(A) a hyperbola

(B) a pair of straight lines

(C) a point

(D) none of the foregoing curves

##### Source of the problem

##### Key Competency

### Coordinate Geometry

##### Difficulty Level

##### Suggested Book

Excursion Of Mathematics

## Start with hints

It cannot be a straight line because Straight line are linear equation of the form. \(ax+by+c=0\).

So it may be a circle or a point if radius is zero, But when we generalized it to the standard form of circle we get negative radius. so it cant be either of this one.

Hint: Stanrd form of circle is $(x-h)^{2}+(y-k)^{2}=r^{2}$.

Now lets try to factorize to find the product of two linear equations so as we can verify the pair of straight line.

$a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$

This equation represents two straight lines, if $\Delta=a b c+2 f g h-a t^{2}-b g^{2}-c h^{2}=0$

or $\left|\begin{array}{lll}{a} & {h} & {g} \\ {h} & {b} & {f} \\ {g} & {f} & {c}\end{array}\right|=0$.

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