# Coordinate Geometry - Class 10 - Mathematics

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## Extra Questions - Coordinate Geometry | NCERT | Mathematics | Class 10

Write the locus of a point whose sum of distances from the coordinate axes is unity.

Let's consider a point $P(x, y)$ such that the **sum of its distances from the coordinate axes (x-axis and y-axis) is 1**. The distance from the x-axis to the point is the absolute value of the y-coordinate, $|y|$, and the distance from the y-axis to the point is the absolute value of the x-coordinate, $|x|$. Therefore, the sum of these distances can be expressed as:

$$ |x| + |y| = 1 $$

This equation represents the locus of points where their combined absolute distances from the two axes equals unity. To more clearly visualize the locus:

When $x \geq 0$ and $y \geq 0$, our equation simplifies to $x + y = 1$.

When $x \geq 0$ and $y < 0$, the equation becomes $x - y = 1$.

When $x < 0$ and $y \geq 0$, it is $-x + y = 1$.

When $x < 0$ and $y < 0$, we simplify to $-x - y = 1$.

All these conditions together describe a figure composed of four line segments, each lying in one quadrant of the Cartesian plane, which ends up forming the edges of a **square** with vertices at $(1,0)$, $(0,1)$, $(-1,0)$, and $(0,-1)$.

Hence, the **locus of the point $P(x, y)$ is a square** defined by the region where $|x| + |y| = 1$.

If $PM$ is the perpendicular from $P(2,3)$ onto the line $x+y=3$, then the coordinates of $M$ are:

A) $(2,1)$

B) $(-1,4)$

C) $(1,2)$

D) $(4,-1)$

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Consider the point $A \equiv (3,4)$ and $B \equiv (7,13)$. If 'P' be a point on line $y=x$ such that $PA+PB$ is minimum, then the coordinates of 'P' are:

A. $\left(\frac{13}{7}, \frac{13}{7}\right)$

B. $\left(\frac{23}{7}, \frac{23}{7}\right)$

C. $\left(\frac{31}{7}, \frac{31}{7}\right)$

D. $\left(\frac{33}{7}, \frac{33}{7}\right)$

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The perpendicular bisector of a line segment AB passes through the origin. If the coordinates of $A$ are $(-2,0)$, then what is the distance of point $B$ from the origin?

A) 2 units

B) -2 units

C) 3 units

D) 4 units

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A line meets $x$-axis and $y$-axis at $A$ and $B$ respectively and $O$ is the origin.

Column I | Column II | Equation of $AB$ | Column | Area of $\Delta C$ |
---|---|---|---|---|

(I) | Centroid $\triangle OAB$ is $(1,2)$ | (i) | $2x+y=2$ | (P) |

(II) | Circumcenter of $\triangle OAB$ is $(1,2)$ | (ii) | $3x+4y=12$ | (Q) |

(III) | Distance of the orthocentre of $\triangle OAB$ from $A$ and $B$ is 1 and 2 respectively | (iii) | $2x+y=6$ | (R) |

(IV) $\quad-2-10$ | Incenter of $\triangle OAB$ is $(1,1)$ | (iv) | $2x+y=4$ | (S) |

Which of the following is correct combination? A (I), (i), (R) B (I), (ii), (P) C (I), (iii), (Q) D (I), (iv), (S)