Chapter 7 · Coordinate Geometry

NCERT Exemplar Class 10 Maths Exercise 7.3

All 20 questions of NCERT Exemplar Class 10 Maths Exercise 7.3 (Coordinate Geometry), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Name ​the type ​of ​triangle ​formed by ​the points $A(-5,6)$, $B(-4,-2)$ and $C(7,5)$.
   Show answer

   All three sides are unequal and no Pythagorean relation holds, so $△ABC$ is a scalene triangle (and not right-angled).

   Step-by-step solution →
2. Q2
   Find the ​points ​on the $x$-axis ​which ​are at a distance of $2\sqrt{5}$ from the point $(7,-4)$. ​How ​many ​such ​points are there?
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   The points are (9, 0) and (5, 0). There are two such points.

   Step-by-step solution →
3. Q3
   What type ​of ​a quadrilateral do ​the points $A(2,-2)$, $B(7,3)$, $C(11,-1)$ ​and $D(6,-6)$ taken in ​that ​order, ​form?
   Show answer

   $ABCD$ is a rectangle: opposite sides $AB=CD=5\sqrt{2}$ and $BC=DA=4\sqrt{2}$, diagonals $AC=BD=\sqrt{82}$; the adjacent sides are unequal, so it is not a square.

   Step-by-step solution →
4. Q4
   Find ​the value of $a$, if ​the distance ​between the ​points $A(-3,-14)$ and $B(a,-5)$ is $9$ ​units.
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   $a=-3$.

   Step-by-step solution →
5. Q5
   Find a ​point which ​is ​equidistant ​from the ​points $A(-5,4)$ and $B(-1,6)$. ​How ​many such points are there?
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   $(-3,5)$ is one such point (the midpoint of $AB$); there are infinitely many, all lying on the perpendicular bisector of $AB$.

   Step-by-step solution →
6. Q6
   Find ​the coordinates of the point $Q$ on the $x$-axis which lies on the perpendicular bisector of ​the line ​segment joining the points $A(-5,-2)$ and $B(4,-2)$. Name the ​type ​of triangle formed by ​the ​points $Q$, $A$ and $B$.
   Show answer

   $Q=\left(-\frac{1}{2},0\right)$; triangle $QAB$ is isosceles since $QA=QB$.

   Step-by-step solution →
7. Q7
   Find the ​value ​of $m$ ​if ​the ​points $(5,1)$, $(-2,-3)$ and $(8,2m)$ are ​collinear.
   Show answer

   $m=\frac{19}{14}$

   Step-by-step solution →
8. Q8
   If ​the point A $(2,-4)$ is equidistant ​from ​P $(3,8)$ ​and Q $(-10,y)$, find the values of $y$. ​Also find ​distance PQ.
   Show answer

   $y=-3$ or $y=-5$. Then $PQ=\sqrt{290}$ (for $y=-3$) or $PQ=13\sqrt{2}$ (for $y=-5$).

   Step-by-step solution →
9. Q9
   Find the ​area ​of ​the ​triangle ​whose vertices are $(-8,4)$, $(-6,6)$ and $(-3,9)$.
   Show answer

   The area is $0$ square units; the three points $(-8,4)$, $(-6,6)$, $(-3,9)$ are collinear.

   Step-by-step solution →
10. Q10
    In what ratio does the ​x-axis divide ​the ​line segment joining the points $(-4,-6)$ and $(-1,7)$? ​Find the ​coordinates ​of the point of ​division.
    Show answer

    The x-axis divides the segment in the ratio $6:7$; the point of division is $\left(-\frac{34}{13},0\right)$.

    Step-by-step solution →
11. Q11
    Find ​the ​ratio ​in which the point $P\left(\frac{3}{4},\frac{5}{12}\right)$ divides ​the line segment joining the points $A\left(\frac{1}{2},\frac{3}{2}\right)$ and $B(2,-5)$.
    Show answer

    $P$ divides $AB$ in the ratio $1:5$ (from $A$ to $B$).

    Step-by-step solution →
12. Q12
    If $P(9a-2,-b)$ divides line segment ​joining $A(3a+1,-3)$ ​and $B(8a,5)$ in ​the ratio $3:1$, ​find the values ​of $a$ ​and $b$.
    Show answer

    $a=1$ and $b=-3$.

    Step-by-step solution →
13. Q13
    If $(a,b)$ ​is the ​mid-point of ​the line segment joining the ​points $A(10,-6)$ ​and $B(k,4)$ ​and $a-2b=18$, ​find ​the value of $k$ and the distance $AB$.
    Show answer

    $k=22$ and $AB=\sqrt{244}=2\sqrt{61}\approx 15.62$ units.

    Step-by-step solution →
14. Q14
    The centre of a circle ​is $(2a,a-7)$. Find the values of $a$ if the circle ​passes ​through the ​point $(11,-9)$ and ​has diameter $10\sqrt{2}$ ​units.
    Show answer

    $a=3$ or $a=5$.

    Step-by-step solution →
15. Q15
    The line segment ​joining the points A(3, 2) and B(5, ​1) is divided ​at the point P in the ​ratio 1:2 and it lies on the line $3x-18y+k=0$. Find ​the ​value ​of k.
    Show answer

    $P=\left(\frac{11}{3},\frac{5}{3}\right)$ and substituting into the line gives $k=19$.

    Step-by-step solution →
16. Q16
    If $D\left(\frac{-1}{2},\frac{5}{2}\right)$, $E(7,3)$ ​and $F\left(\frac{7}{2},\frac{7}{2}\right)$ are the ​midpoints of ​sides of $△ABC$, find ​the area ​of the $△ABC$.
    Show answer

    Area of $△ABC=11$ square units.

    Step-by-step solution →
17. Q17
    The ​points $A(2,9)$, $B(a,5)$ and $C(5,5)$ ​are the vertices of a triangle $ABC$ ​right angled at $B$. Find ​the values of $a$ ​and hence the ​area of $△ABC$.
    Show answer

    a = 2 only; Area = 6 square units

    Step-by-step solution →
18. Q18
    Find the coordinates of ​the ​point $R$ on the ​line segment joining the ​points $P(-1,3)$ ​and $Q(2,5)$ ​such that $PR=\frac{3}{5}PQ$.
    Show answer

    $R=\left(\frac{4}{5},\frac{21}{5}\right)$.

    Step-by-step solution →
19. Q19
    Find ​the values ​of $k$ ​if ​the ​points $A(k+1,2k)$, $B(3k,2k+3)$ and $C(5k-1,5k)$ ​are collinear.
    Show answer

    $k=\frac{1}{2}$ or $k=2$.

    Step-by-step solution →
20. Q20
    Find ​the ratio ​in ​which the line $2x+3y-5=0$ divides the line segment ​joining the points $(8,-9)$ and $(2,1)$. Also find the coordinates ​of ​the point of ​division.
    Show answer

    The line divides the segment in the ratio $8:1$, and the point of division is $\left(\frac{8}{3},-\frac{1}{9}\right)$.

    Step-by-step solution →