Question
Find the point on the y-axis which is equidistant from the points A(6, 5) and B(-4, 3).
✓
Answer
Let the required point be C(0, y)
Then, we have
$AC = BC$
$\Rightarrow AC^2 = BC^2$
$\Rightarrow (6 - 0)^2 + (5 - y)^2 = (-4 - 0)^2 + (3 - y)^2$
$\Rightarrow (6)^2 + (25 + y^2 - 10y) = (-4)^2 + (9 + y^2 - 6y)$
$\Rightarrow 36 + 25 + y^2 - 10y = 16 + 9 + y^2 - 6y$
$\Rightarrow 61 - 10y = 25 - 6y$
$\Rightarrow 4y = 36$
$\Rightarrow y = 9$
Hence, the required point is C(0, 9).
Then, we have
$AC = BC$
$\Rightarrow AC^2 = BC^2$
$\Rightarrow (6 - 0)^2 + (5 - y)^2 = (-4 - 0)^2 + (3 - y)^2$
$\Rightarrow (6)^2 + (25 + y^2 - 10y) = (-4)^2 + (9 + y^2 - 6y)$
$\Rightarrow 36 + 25 + y^2 - 10y = 16 + 9 + y^2 - 6y$
$\Rightarrow 61 - 10y = 25 - 6y$
$\Rightarrow 4y = 36$
$\Rightarrow y = 9$
Hence, the required point is C(0, 9).
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the coordinate of the point which divides the join of A(-5, 11) and B(4, -7) in the ratio 7 : 2.
A right angled triangle whose sides are 3cm, 4cm and 5cm is revolved about the sides containing the right angle in two ways. Find the difference in volumes of the two cones so formed. Also, find their curved surfaces.
A right angled triangle with sides 3cm and 4cm is revolved around its hypotenuse. Find the volume of the double cone thus generated.
Prove that the points (3, 0), (6, 4) and (-1, 3) are vertices of a right-angled isosceles triangle.
A survey regarding the height (in cm) of 51 girls of class X of a school was conducted and the following data was obtained.
Find the median height.
|
Hight in cm
|
Number of girls
|
|
Less than 140
|
4
|
|
Less than 145
|
11
|
|
Less than 150
|
29
|
|
Less than 155
|
40
|
|
Less than 160
|
46
|
|
Less than 165
|
51
|
In Fig. CM and RN are respectively the medians of $\triangle$ABC and $\triangle$PQR. If $\triangle$ABC $\sim\triangle$PQR, prove that:
- $\triangle$AMC $\sim\triangle$PNR
- $\frac{C M}{R N}=\frac{A B}{P Q}$
- $\triangle$CMB $\sim\triangle$RNQ
The median of the following data is $525.$ Find the values of $x$ and $y,$ if the total frequency is $100.$
→| Class interval | Frequency |
| $0-100$ | $2$ |
| $100-200$ | $5$ |
| $200-300$ | $x$ |
| $300-400$ | $12$ |
| $400-500$ | $17$ |
| $500-600$ | $20$ |
| $600-700$ | $y$ |
| $700-800$ | $9$ |
| $800-900$ | $7$ |
| $900-1000$ | $4$ |
A pole $6 m$ high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point $P$ on the ground is $60^{\circ}$ and the angle of depression of the point $P$ from the top of the tower is $45^{\circ}$. Find the height of the tower and the distance of point $P$ from the foot of the tower. $($Use $\sqrt{3}=1.73)$
→Solve the following system of equations by the method of cross-multiplication:
→$mx - ny = m^2 + n^2$
$x + y = 2m$
Find the value of p for which quadratic equation $(p + 1)x^2 - 6(p + 1)x + 3(p + 9) = 0$, $\text{p}\neq-1$ has equal roots. Hence, find the roots of the equation.
→