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Q8:
Find the values of $y$ for which the distance between the points P(2, – 3) and Q(10, $y$) is 10 units.
Solution :
Given:
The coordinates of point $P$ are $(x_1, y_1) = (2, -3)$.
The coordinates of point $Q$ are $(x_2, y_2) = (10, y)$.
The distance between points $P$ and $Q$ is $d = 10$ units.
To find:
The value(s) of $y$.
Step 1: Applying the Distance Formula
The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a Cartesian plane is given by the formula:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
[By the Distance Formula in Coordinate Geometry]
Step 2: Substituting the Given Values
Substitute $d = 10$, $x_1 = 2$, $y_1 = -3$, $x_2 = 10$, and $y_2 = y$ into the formula:
$10 = \sqrt{(10 - 2)^2 + (y - (-3))^2}$
$10 = \sqrt{(8)^2 + (y + 3)^2}$
Step 3: Squaring Both Sides to Eliminate the Square Root
To solve for $y$, we square both sides of the equation:
$(10)^2 = (\sqrt{(8)^2 + (y + 3)^2})^2$
$100 = 8^2 + (y + 3)^2$
$100 = 64 + (y + 3)^2$
Step 4: Simplifying the Equation
Subtract 64 from both sides:
$100 - 64 = (y + 3)^2$
$36 = (y + 3)^2$
Step 5: Solving for $y$
Take the square root of both sides, remembering to include both positive and negative roots:
$\pm \sqrt{36} = y + 3$
$\pm 6 = y + 3$
This yields two distinct linear equations:
Case 1: $6 = y + 3$
$y = 6 - 3$
$y = 3$
Case 2: $-6 = y + 3$
$y = -6 - 3$
$y = -9$
Conclusion:
The possible values for $y$ that satisfy the condition are $3$ and $-9$.
Final Answer: The values of $y$ are $3$ and $-9$.
More Questions from Class 10 Mathematics Coordinate geometry EXERCISE 7.1
- Q1(i): Find the distance between the following pairs of points : (i) (2, 3), (4, 1)
- Q1(ii): Find the distance between the following pairs of points : (ii) (– 5, 7), (– 1, 3)
- Q1(iii): Find the distance between the following pairs of points : (iii) ($a$, $b$), (– $a$, – $b$)
- Q10: Find a relation between $x$ and $y$ such that the point ($x$, $y$) is equidistant from the point (3, 6) and (– 3, 4).
- Q2: Find the distance between the points (0, 0) and (36, 15). Can you now find the distance between the two towns A and B.
- Q3: Determine if the points (1, 5), (2, 3) and (– 2, – 11) are collinear.
- Q4: Check whether (5, – 2), (6, 4) and (7, – 2) are the vertices of an isosceles triangle.
- Q5: In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 7.8. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees. Using distance formula, find which of them is correct.
- Q6(i): Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer: (i) (– 1, – 2), (1, 0), (– 1, 2), (– 3, 0)
- Q6(ii): Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer: (ii) (–3, 5), (3, 1), (0, 3), (–1, – 4)
- Q6(iii): Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer: (iii) (4, 5), (7, 6), (4, 3), (1, 2)
- Q7: Find the point on the x-axis which is equidistant from (2, –5) and (–2, 9).
- Q9: If Q(0, 1) is equidistant from P(5, –3) and R($x$, 6), find the values of $x$. Also find the distances QR and PR.
CBSE Solutions for Class 10 Mathematics Coordinate geometry
Chapters in CBSE - Class 10 Mathematics
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