R.S. Aggarwal solutions for Mathematics [English] Class 10 chapter 6 - Coordinate Geometry [Latest edition]

Find the distance between the points:

A(9, 3) and B(15, 11)

Find the distance between the points:

A(7, –4) and B(–5, 1)

Find the distance between the points:

A(–6, –4) and B(9, –12)

Find the distance between the points:

A(1, –3) and B(4, –6)

Find the distance between the points:

P(a + b, a – b) and Q(a – b, a + b)

Find the distance between the points:

P(a sin α, a cos α) and Q(a cos α, – a sin α)

Find the distance of the following points from the origin:

A(5, –12)

Find the distance of the following points from the origin:

B(–5, 5)

Find the distance of the following points from the origin:

C(–4, –6)

Find all possible values of x for which the distance between the points A(x, –1) and B(5, 3) is 5 units.

Find all possible values of y for which the distance between the points A(2, –3) and B(10, y) is 10 units.

Find the values of x for which the distance between the points P(x, 4) and Q(9, 10) is 10 units.

If the point A(x, 2) is equidistant from the points B(8, –2) and C(2, –2), find the value of x. Also, find the length of AB.

If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find the value of p. Also, find the length of AB.

Find the point on the x-axis which is equidistant from (2, –5) and (–2, 9).

Find the points on the x-axis, each of which is at a distance of 10 units from the point A(11, –8).

Find the points on the y-axis which is equidistant from the points A(6, 5) and B(–4, 3). 

If the points P(x, y) is equidistant from the points A(5, 1)and B(–1, 5), prove that 3x = 2y.

If P(x, y) is point equidistant from the points A(6, –1) and B(2, 3), show that x – y = 3.

Find the coordinates of the point equidistant from three given points A(5, 3), B(5, –5) and C(1, –5).

If the point A(4, 3) and B(x, 5) lies on a circle with the centre O(2, 3), find the value of x.

HINT: OA² = OB².

If the point C(–2, 3) is equidistant from the points A(3, –1) and B(x, 8), find the values of x. Also, find the distance between BC.

If the point P(2, 2) is equidistant from the points A(–2, k) and B(–2k, –3), find k. Also, find the length of AP.

If the point (x, y) is equidistant form the points (a + b, b – a) and (a – b, a + b), prove that bx = ay.

Using the distance formula, show that the given points are collinear:  

 (1, –1), (5, 2) and (9, 5)

Using the distance formula, show that the given points are collinear:

(6, 9), (0, 1) and (–6, –7)

Using the distance formula, show that the given points are collinear:

(–1, –1), (2, 3) and (8, 11)

Using the distance formula, show that the given points are collinear:

(–2, 5), (0, 1) and (2, –3)

Prove that the points A(7, 10), B(–2, 5) and C(3, –4) are the vertices of an isosceles right triangle.

Show that the points A(3, 0), B(6, 4) and C(–1, 3) are the vertices of an isosceles right triangle.

If A(5, 2), B(2, –2) and C(–2, t) are the vertices of a right triangle with ∠B = 90°, then find the value of t.

Prove that the points A(2, 4), B(2, 6) and `C(2 + sqrt(3), 5)` are the vertices of an equilateral triangle.

Show that the points (–3, –3), (3, 3) and `(-3sqrt(3), 3sqrt(3))` are the vertices of an equilateral triangle.

Show that the points A(–5, 6), B(3, 0) and C(9, 8) are the vertices of an isosceles right-angled triangle. Calculate its area.

Show that the points O(0, 0), `A(3, sqrt(3))` and `B(3, -sqrt(3))` are the vertices of an equilateral triangle. Find the area of this triangle.

Show that the following points are the vertices of a square:

A(3, 2), B(0, 5), C(–3, 2) and D(0, –1)

Show that the following points are the vertices of a square:

A(6, 2), B(2, 1), C(1, 5) and D(5, 6)

Show that the following points are the vertices of a square:

A(0, –2), B(3, 1), C(0, 4) and D(–3, 1)

Show that the points A(–3, 2), B(–5, –5), C(2, –3) and D(4, 4) are the vertices of a rhombus. Find the area of this rhombus.

HINT: Area of a rhombus = `1/2` × (product of its diagonals).

Show that the points A(3, 0), B(4, 5), C(–1, 4) and D(–2, –1) are the vertices of a rhombus. Find its area.

Show that the points A(6, 1), B(8, 2), C(9, 4) and D(7, 3) are the vertices of a rhombus. Find its area.

Show that the points A(2, 1), B(5, 2), C(6, 4) and D(3, 3) are the angular points of a parallelogram. Is this figure a rectangle?

Show hat A(1, 2), B(4, 3), C(6, 6) and D(3, 5) are the vertices of a parallelogram. Show that ABCD is not a rectangle.

Prove that the points A(–4, –1), B(–2, –4), C(4, 0) and D(2, 3) are the vertices of a rectangle.

Show that the following points are the vertices of a rectangle:

A(2, –2), B(14, 10), C(11, 13) and D(–1,1)

Show that the following points are the vertices of a rectangle:

A(0, –4), B(6, 2), C(3, 5) and D(–3, –1)

Show that ΔABC with vertices A(–2, 0), B(0, 2) and C(2, 0) is similar to ΔDEF with vertices D(–4, 0), E(0, 4) and F(4, 0).

Find the coordinates of the point which divides the join of A(–1, 7) and B(4, –3) in the ratio 2 : 3.

Find the coordinates of the point which divides the join of A(–5, 11) and B(4, –7) in the ratio 7 : 2.

Find the coordinates of the points of trisection of the line segment joining the points A(7, –2) and B(1, –5).

If the coordinates of points A and B are (–2, –2) and (2, –4) respectively, find the coordinates of the point P such that AP = `3/7` AB, where P lies on the line segment AB.

Point A lies on the line segment PQ joining P(6, –6) and Q(–4, –1) in such a way that `(PA)/(PQ) = 2/5`. If the point A also lies on the line 3x + k(y + 1) = 0, find the value of k.

Points P, Q, R and S divide the line segment joining the points A(1, 2) and B(6, 7) in five equal parts. Find the coordinates of the points P, Q and R.

Points P, Q and R in that order are dividing a line segment joining A(1, 6) and B(5, –2) in four equal parts. Find the coordinates of P, Q and R.

The line segment joining the points A(3, –4) and B(1, 2) is trisected at the points P(p, –2) and `Q(5/3, q)`. Find the values of p and q.

Find the coordinates of the midpoint of the line segment joining

A(3, 0) and B(–5, 4)

Find the coordinates of the midpoint of the line segment joining 

P(–11, –8) and Q(8, –2)

If (2, p) is the midpoint of the line segment joining the points A(6, –5) and B(–2, 11), find the value of p.

The midpoint of the line segment joining A(2a, 4) and B(–2, 3b) is C(1, 2a + 1). Find the values of a and b.

The line segment joining A(–2, 9) and B(6, 3) is a diameter of a circle with centre C. Find the coordinates of C.

Find the coordinates of a point A, where AB is the diameter of circle whose centre is (2, –3) and B is (1, 4).

In what ratio does the point P(2, 5) divide the join of A(8, 2) and B(–6, 9)?

Find the ratio in which the poin `P(3/4, 5/12)` divides the line segment joining the points `A(1/2, 3/2)` and B(2, –5).

Find the ratio in which the point P(m, 6) divides the join of A(–4, 3) and B(2, 8). Also, find the value of m. 

Find the ratio in which the point (–3, k) divides the join of A(–5, –4) and B(–2, 3). Also, find the value of k.

In what ratio is the line segment joining A(2, –3) and B(5, 6) divided by the x-axis? Also, find the coordinates of the point of division.

In what ratio is the line segment joining the points A(–2, –3) and B(3, 7) divided by the y-axis? Also, find the coordinates of the point of division.

In what ratio does the line x – y – 2 = 0 divide the line segment joining the points A(3, –1) and B(8, 9)? 

Find the lengths of the medians of a ΔABC whose vertices are A(0, –1), B(2, 1) and C(0, 3).

Find the centroid of ΔABC whose vertices are A(–1, 0), B(5, –2) and C(8, 2).

If G(–2, 1) is the centroid of a ΔABC and two of its vertices are A(1, –6) and B(–5, 2), find the third vertex of the triangle.

Find the third vertex of a ΔABC if two of its vertices are B(–3, 1) and C(0, –2) and its centroid is at the origin.

Show that the points A(3, 1), B(0, –2), C(1, 1) and D(4, 4) are the vertices of a parallelogram ABCD.

If the points P(a, –11), Q(5, b), R(2, 15) and S(1, 1) are the vertices of a parallelogram PQRS, find the values of a and b.

If three consecutive vertices of a parallelogram are (1, –2), (3, 6) and (5, 10), find its fourth vertex.

In what ratio does y-axis divide the line segment joining the points (–4, 7) and (3, –7)?

If the point `P (1/2, y)` lies on the line segment joining the points A(3, –5) and B(–7, 9) then find the ratio in which P divides AB. Also, find the value of y.

Find the ratio in which the line segment joining the points A(3, –3) and B(–2, 7) is divided by x-axis. Also, find the point of division.

The base QR of an equilateral triangle PQR lies on x-axis. The coordinates of the point Q are (–4, 0) and origin is the midpoint of the base. Find the coordinates of the points P and R.

The base BC of an equilateral triangle ABC lies on y-axis. The coordinates of point C are (0, −3). The origin is the midpoint of the base. Find the coordinates of the points A and B. Also, find the coordinates of another point D such that ABCD is a rhombus.

Find the ratio in which the point P(–1, y) lying on the line segment joining points A(–3, 10) and B(6, –8) divides it. Also, find the value of y.

ABCD is a rectangle formed by the points A(–1, –1), B(–1, 4), C(5, 4) and D(5, –1). If P, Q, R and S be the midpoints of AB, BC, CD and DA respectively, show that PQRS is a rhombus.

The midpoint P of the line segment joining the points A(–10, 4) and B(–2, 0) lies on the line segment joining the points C(–9, –4) and D(–4, y). Find the ratio in which P divides CD. Also, find the value of y.

A line intersects the y-axis and x-axis at the points P and Q respectively. If (2, –5) is the mid-point of PQ, then find the coordinates of P and Q.

In what ratio does the point `(24/11, y)` divide the line segment joining the points P(2, –2) and Q(3, 7)? Also, find the value of y.

The midpoints of the sides BC, CA and AB of a ΔАBC are D(3, 4), E(8, 9) and F(6, 7) respectively. Find the coordinates of the vertices of the triangle.

If two adjacent vertices of a parallelogram are (3, 2) and (−1, 0) and the diagonals intersect at (2, −5), then find the coordinates of the other two vertices.

Points A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD. Find the values of a and b.

The line segment joining the points A(2, 1) and B(5, −8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by 2x − y + k = 0, find the value of k.

Find the ratio in which y-axis divides the line segment joining the points A(5, –6) and B(–1, –4). Also find the coordinates of the point of division.

If the mid-point of the line segment joining the points A(3, 4) and B(k, 6) is P(x, y) and x + y – 10 = 0, find the value of k.

Find the area of ΔABC whose vertices are A(1, 2) B(–2, 3) and C(–3, –4).

Find the area of ΔABC whose vertices are A(–5, 7), B(–4, –5) and C(4, 5).

Find the area of ΔABC whose vertices are A(3, 8), B(–4, 2) and C(5, –1).

Find the area of ΔABC whose vertices are A(10, –6), B(2, 5) and C(–1, 3).

Find the area of quadrilateral ABCD whose vertices are A(3, –1), B(9, –5), C(14, 0) and D(9, 19).

Find the area of quadrilateral PQRS whose vertices are P(–5, –3), Q(–4, –6), R(2, –3) and S(1, 2).

Find the area of quadrilateral ABCD whose vertices are A(–3, –1), B(–2, –4), C(4, –1) and D(3, 4).

If A(–7, 5), В(–6, –7), С(–3, –8) and D(2, 3) are the vertices of a quadrilateral ABCD, then find the area of the quadrilateral.

Find the area of the triangle formed by joining the midpoints of the sides of the triangle whose vertices are A(2, 1), B(4, 3) and C(2, 5).

A(7, –3), B(5, 3) and C(3, –1) are the vertices of a ΔABC and AD is its median. Prove that the median AD divides ΔABC into two triangles of equal areas. 

Find the area of ΔABC with A(1, –4) and midpoints of sides through A being (2, –1) and (0, –1).

A(6, 1), B(8, 2) and C(9, 4) are the vertices of a parallelogram ABCD. If E is the midpoint of DC, find the area of ΔADE.

If the vertices of ΔABC be A(1, –3), B(4, p) and C(–9, 7) and its area is 15 square units, find the values of p.

The area of a triangle is 5 sq units. Two of its vertices are (2, 1) and (3, –2). If the third vertex is `(7/2, y)`, find the value of y.

Find the value of k so that the area of the triangle with vertices A(k + 1, 1), B(4, –3) and C(7, –k) is 6 square units. 

For what value of k(k > 0) is the area of the triangle with vertices (–2, 5), (k, –4) and (2k + 1, 10) equal to 53 square units?

Show that the following points are collinear:

A(2, –2), B(–3, 8) and C(–1, 4)

Show that the following points are collinear:

A(–5, 1), B(5, 5) and C(10, 7)

Show that the following points are collinear:

A(5, 1), B(1, –1) and C(11, 4)

Show that the following points are collinear:

A(8, 1), B(3, –4) and C(2, –5)

Find the value of x for which the points A(x, 2), B(–3, –4) and C(7, –5) are collinear.

For what value of x are the points A(−3, 12), B(7, 6) and C(x, 9) collinear.

Find the value of p for which the points A(–5, 1), B(1, p) and C(4, –2) are collinear.

Find the value of y for which the points A(–3, 9), B(2, y) and C(4, –5) are collinear.

For what values of k are the points A(8, 1), B(3, –2k) and C(k, –5) collinear?

Find a relation between x and y, if the points A(x, y), B(1, 2) and C(7, 0) are collinear.

Find a relation between x and y, if the points A(x, y), B(–5, 7) and C(–4, 5) are collinear.

Prove that the points A(a, 0), B(0, b) and C(1, 1) are collinear, if `(1/a + 1/b) = 1`.

If the points P(–3, 9), Q(a, b) and R(4, –5) are collinear and a + b = 1, find the values of a and b.

Find the area of ΔABC with vertices A(0, –1), B(2, 1) and C(0, 3). Also, find the area of the triangle formed by joining the midpoints of its sides. Show that the ratio of the areas of two triangles is 4 : 1.

If a ≠ b ≠ 0, prove that the points (a, a²), (b, b²) (0, 0) will not be collinear.

If the area of the triangle with vertices (x, 3), (4, 4) and (3, 5) is 4 square units, find x.

Points A(–1, y) and B(5, 7) lie on a circle with centre O(2, –3y). Find the values of y.

If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p.

ABCD is a rectangle whose three vertices are B(4, 0), C(4, 3) and D(0, 3). Find the length of one of its diagonals.

If the point P(k – 1, 2) is equidistant from the points A(3, k) and B(k, 5), find the value of k.

Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, –3). Also, find the value of x.

Prove that the diagonals of a rectangle ABCD with vertices A(2, –1), B(5, –1), C(5, 6) and D(2, 6) are equal and bisect each other.

Find the lengths of the medians of a ∆ABC whose vertices are A(7, –3), B(5, 3) and C(3, –1).

If the point C(k, 4) divides the join of A(2, 6) and B(5, 1) in the ratio 2 : 3 then find the value of k. 

Find the point on x-axis which is equidistant from points A(–1, 0) and B(5, 0).

Find the distance between the points `A((-8)/5, 2)` and `B(2/5, 2)`.

Find the value of a, so that the point (3, a) lies on the line represented by 2x – 3y = 5.

If the points A(4, 3) and B(x, 5) lie on the circle with centre O(2, 3), find the value of x.

If P(x, y) is equidistant from the points A(7, 1) and B(3, 5), find the relation between x and y.

If the centroid of ΔABC having vertices A(a, b), B(b, c) and C(c, a) is the origin, then find the value of (a + b + c).

Find the centroid of ΔABC whose vertices are A(2, 2), B(–4, –4) and C(5, –8).

In what ratio does the point C(4, 5) divide the join of A(2, 3) and B(7, 8)?

 If the points A(2, 3), B(4, k) and C(6, –3) are collinear, find the value of k.

The distance of the point P(–6, 8) from the origin is ______.

The distance of the point (–3, 4) from the x-axis is ______. 

The point on the x-axis which is equidistant from points (–1, 0) and (5, 0) is ______.

If the point C(k, 4) divides the join of the points A(2, 6) and B(5, 1) in the ratio 2 : 3 then the value of k is ______.

The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is ______.

If A(1, 3), B(–1, 2), C(2, 5) and D(x, 4) are the vertices of a ||gm ABCD then the value of x is ______.

If the points A(x, 2), B(–3, –4) and C(7, –5) are collinear, then the value of x is ______.

The area of a triangle with vertices A(5, 0), B(8, 0) and C(8, 4) in square units is ______.

The area of ΔABC with vertices A(a, 0), O(0, 0) and B(0, b) in square units is ______.

If `P(a/2, 4)` is the mid-point of the line segment joining the points A(–6, 5) and B(–2, 3), then the value of a is ______.

ABCD is a rectangle whose three vertices are B(4, 0), C(4, 3) and D(0, 3). The length of one of its diagonals is ______.

The coordinates of the point P dividing the line segment joining the points A(1, 3) and B(4, 6) in the ratio 2 : 1 is ______.

If the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its centre are (–2, 5) then the coordinates of the other end of the diameter are ______.

In the given figure P(5, –3) and Q(3, y) are the points of trisection of the line segment joining A(7, –2) and B(1, –5). Then, y equals

The midpoint of segment AB is P(0, 4). If the coordinates of B are (–2, 3) then the coordinates of A are ______.

The point P which divides the line segment joining the points A(2, –5) and B(5, 2) in the ratio 2 : 3 lies in the quadrant

If A(–6, 7) and B(–1, –5) are two given points then the distance 2AB is ______.

Which point on x-axis is equidistant from the points A(7, 6) and B(–3, 4)?

Distance of point P(3, 4) from x-axis is ______.

In what ratio does the x-axis divide the join of A(2, –3) and B(5, 6)?

In what ratio does the y-axis divide the join of P(–4, 2) and Q(8, 3)?

If P(–1, 1) is the midpoint of the line segment joining A(–3, b) and B(1, b + 4) then b = ?

The line 2x + y – 4 = 0 divides the line segment joining A(2, –2) and B(3, 7) in the ratio