Find a relation between x and y such that the point P(x, y) is equidistant from the points A(5, 3) and B(1, 7).

Find a relation between x and y such that the point P(x, y) is equidistant from the

 points A(5, 3) and B(1, 7)

Given

P(x, y) is equidistant from A and B, 

 PA = PB

squaring both sides

PA² = PB²

Using distance formula to find PA and PB

(x - 5)² + (y - 3)² = (x - 1)² + (y - 7

Expand both sides using identity

x² - 10x + 25 + y² - 6y + 9 = x² - 2x + 1 + y² - 14y + 49

-10x - 6y + 34 = -2x - 14y + 50

-10x + 2x - 6y + 14y = 50 - 34

-8x + 8y = 16

or

-x + y = 2

or 

y = x + 2

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cbse 10th maths 2025 2026 old board exam question paper coordinate geometry distance formula

real numbers HCF LCM cbse 10th maths ( part 3)

 real numbers HCF LCM cbse 10th maths ( part 3) 

link for real numbers, hcf lcm (part 1)

click here for real numbers HCF LCM (part2)

Find the smallest number which is divisible by both 644 and 462.

solution 

2.

Two numbers are in the ratio 4: 5 and their HCF is 11. Find the LCM of these numbers.

solution

3.

Given that HCF (306, 1314) = 18,
find LCM of (306, 1314).

solution 

real Numbers H.C.F and L.C.M cbse 10th maths old board exam question paper sample question paper sqp

 real numbers , hcf , lcm ... (part2)

link for real numbers, hcf lcm (part 1)

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prove that{  [  2 -  √(3) ] / 5 } is irrational

given that √(3) is irrational

solution 

2.

Find the H.C.F and L.C.M of 480 and 720 using the Prime factorisation method.

solution 

3.

There is a circular path around a sports  field.Three cyclists start from the same point and at the same time and go in the same direction.If they take  30 minutes,  40 minutes and 48 minutes  respectively to  complete one  round of the field, after how  many  minutes will they meet  again at the starting point?

solution 

4.

Two alarm clocks ring  their alarms at regular intervals of 20 minutes and 25 minutes respectively. If they first beep together at 12 noon, at what time will they beep again together next time?

solution

5.

Teaching Mathematics through activities is a powerful approach that enhances students' understanding and engagement. Keeping this in mind, Ms. Mukta planned a prime number game for class 5 students. She announces the number 2 in her class and asked the first student to multiply it by a prime number and then pass it to second student. Second student also multiplied it by a prime number and passed it to third student. In this way by multiplying to a prime number, the last student got 173250.

solution 

link for real numbers, hcf lcm (part 1)

A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of stream

 A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of stream

x=speed of the stream in km/hr 

speed of boat upstream =(18-x)km/hr

speed of the boat downstrream=(18+x)km/hr

distance=24km

time=distance/speed

t1=24/(18-x)hours upstream

t2=24/(18+x) hours downstream

given t1=t2  + 1

t1 - t2=1

[24 / (18-x)]-[24/(18+x)]=1

48x/[324-(x"2)]=1

(x"2)+48x-324=0

[x-6][x+54]=0

x=6 or x= -54 [reject]

speed of stream =6 km/hr

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 questions in old cbse sample question papers for 10th  mathematics

  National Art convention got registrations from students from all parts of the country, of which 60 are interested in music, 84 are interested in dance and 108 students are interested in handicrafts. For optimum cultural exchange, organisers wish to keep them in minimum number of groups such that each group consists of students interested in the same art form and the number of students in each group is the same. Find the number of students in each group. Find the number of groups in each art form. How many rooms are required if each group will be allotted a room

solution

 

 

if 49x+51y=499 ,51x+49y=501, find the value of x and y

 solution

A had some chocolates, and he divided them into two lots A and B. He sold the first
lot at the rate of ₹2 for 3 chocolates and the second lot at the rate of ₹1 per chocolate, and got a total of ₹400. If he had sold the first lot at the rate of ₹1 per  chocolate, and the second lot at the rate of ₹4 for 5 chocolates, his total collection  would have been ₹460.
Find the total number of chocolates he had.

solution

 

The length of the minute hand of a clock is 6cm. Find the area swept by it when it moves from 7:05 p.m. to 7:40 p.m. 

solution

n the given figure, arcs have been drawn of radius 7cm each with vertices A, B, C
and D of quadrilateral ABCD as centres. Find the area of the shaded region consisting of the four sectors at the four vertices

 

solution

If sin(A+B) =1 and cos(A-B)= √3/2, 0°< A+B ≤ 90° and A> B, then find the
measures of angles A and B

solution

Find an acute angle θ when (cosθ − sin θ)/(cosθ+sin θ) = (1−√3)/(1+√3)

 solution

 

 If the zeroes of the polynomial (x^2) +px +q are double in value to the zeroes of the polynomial 2(x^2 )-5x -3, then find the values of p and q.

 solution

 

 If A+B=pi/4 show that [1+tan A][1+tanB]=2  or  [1+cotA][1+cotB]=2cotAcotB

https://keral2008.blogspot.com/2024/05/if-abpi4-show-that-1tan-a1tanb2-or.html

Two coins are tossed simultaneously. What is the probability of getting
(i) At least one head?
(ii) At most one tail?
(iii) A head and a tail?

 solution

 To fill a swimming pool two pipes are used. If the pipe of larger diameter used for 4 hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool?

 solution

 

The school auditorium was to be constructed to accommodate at least 1500 people. The chairs are to be placed in concentric circular arrangement in such a way that each succeeding circular row has 10 seats more than the previous one.
(i) If the first circular row has 30 seats, how many seats will be there in the 10th row?
(ii) For 1500 seats in the auditorium, how many rows need to be there?

If 1500 seats are to be arranged in the auditorium, how many seats are still left to
be put after 10th row?
(iii) If there were 17 rows in the auditorium, how many seats will be there in the
middle row

solution

 Due to heavy floods in a state, thousands were rendered homeless. 50 schools
collectively decided to provide place and the canvas for 1500 tents and share the
whole expenditure equally. The lower part of each tent is cylindrical with base
radius 2.8 m and height 3.5 m and the upper part is conical with the same base
radius, but of height 2.1 m. If the canvas used to make the tents costs ₹120 per m^2,find the amount shared by each scho15ol to set up the tents

solution

 If tan (A + B) = √3 and tan (A – B) = 1/√3
 0° < A + B < 90°; A > B, find A and B

solution

 Find the value of x if
2 [ cosec30)^2 ] + x  [(sin60)^2] – (3/4)(tan30)^2 = 10 where the angles are in degrees

 solution

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